Photonic Crystals: Fabrication, Band Structure and Applications: Fabrication, Band Structure and Applications [1 ed.] 9781617283406, 9781616689537

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers, Incorporated,

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

PHYSICS RESEARCH AND TECHNOLOGY

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PHOTONIC CRYSTALS: FABRICATION, BAND STRUCTURE AND APPLICATIONS

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PHYSICS RESEARCH AND TECHNOLOGY

PHOTONIC CRYSTALS: FABRICATION, BAND STRUCTURE AND APPLICATIONS

VENLA E. LAINE

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Photonic crystals : fabrication, band structure, and applications / editors, Venla E. Laine. p. cm. Includes index.

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1. Photonic crystals. I. Laine, Venla E. QD924.P465 2010 548'.83--dc22 2010014111

Published by Nova Science Publishers, Inc. † New York

Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

CONTENTS

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Preface

vii

Chapter 1

Fabrication and Applications of Polymer Photonic Crystals Jingxia Wang and Yanlin Song

Chapter 2

Achieving Complete Band Gaps Using Low Refractive Index Material Dahe Liu, Tianrui Zhai and Zhaona Wang

31

Chapter 3

Photonic Crystals for Microwave Applications Yoshihiro Kokubo

67

Chapter 4

Physics of Photonic Crystal Couplers and Their Applications Chih-Hsien Huang, Forest Shih-Sen Chien, Wen-Feng Hsieh and Szu-Cheng Cheng

95

Chapter 5

A Photonic Band Gap in Quasicrystal-Related Structures P.N. Dyachenko and Yu.V. Miklyaev

115

Chapter 6

Spectrum Engineering with One-Dimensional Photonic Crystal Anirudh Banerjee

135

Chapter 7

Transmission through Kerr Media Barriers within Waveguides: Device Applications A.R. McGurn

149

Chapter 8

Influence of the Solid-Liquid Adhesion on the Self-Assembling Properties of Colloidal Particles Edina Rusen, Alexandra Mocanu, Bogdan Marculescu, Corina Andronescu, I.C. Stancu, L.M. Butac, A.Ioncea and I. Antoniac

173

Chapter 9

Optical Logic Gate Based in a Photonic Crystal A. Wirth L. Jr and A.S.B. Sombra

191

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1

vi

Contents

Chapter 10

Two-Dimensional Photonic Crystals with Glide-Plane Symmetry for Integrated Photonics Applications Adam Mock and Ling Lu

209

Chapter 11

Nonlinear Photonic Crystals of Strontium Tetraborate A.S. Aleksandrovsky, A.M. Vyunishev and A.I. Zaitsev

245

Chapter 12

Optical Properties of Self-Assembled Colloidal Photonic Crystals Doped with Citrate-Capped Gold Nanoparticles Marco Cucini, Marina Alloisio, Anna Demartini and Davide Comoretto

277

293

Index

297

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Short Communication Mathematical Representation of the Group Index in a Photonic Crystal M.A. Grado-Caffaro and M. Grado-Caffaro

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PREFACE Photonic crystals are periodic optical nanostructures that are designed to affect the motion of photons in a similar way that periodicity of a semiconductor crystal affects the motion of electrons. Photonic crystals occur in nature and in various forms have been studied scientifically for the last 100 years. This book gathers and presents topical data in the field of photonic crystals including the phenomena in photonic crystals stipulating the presence of omnidirectional band gaps, i.e., overlapping of stop bands in all directions; the physics of photonic crystal couplers and their applications; photonic crystals and their capability to control the propagation and emission of electromagnetic waves and others. According to their periodicity in space, PCs can be categorized mainly into onedimensional (1D), 2D and 3D PCs. All above-mentioned PCs have been fabricated artificially from a variety of materials. To date, the main 3D PCs include opal, inverse opal, woodpile, diamond structure, etc. Among those structures, opal structure comprised of polymer colloidal crystals demonstrates promising advantages over other structures, for example, polymer colloidal is easily processed, and its properties could be easily modulated based on the design of molecular structure. Moreover, polymer colloidal crystal can also act as template for the fabrication of PCs of inorganic, metal and complicate materials, since nanometer sized polymer particles are readily available with a high mono-dispersity or can be synthesized in a controlled way. Plenty of review papers[3-9] have been published about the fabrication of polymer PCs and its special applications. Such as, fabrication of polymer PCs form latex system[3-5] inverse opal structure[6], and micro-phase separation[10]. Polymer PC material is a novel material, and can provide material basis for the development of relative subjects, which covers photo-computing, photo-communication and optic-electric integration. In Chapter 1, the authors review the recent research progress of polymer PCs, mainly concern on the fabrication and application of polymer PCs. Photonic crystals (PCs) were introduced by E. Yablonovitch [1] and S. John [2] in 1987. In 1990, Ho et al. demonstrated theoretically that a diamond structure possesses complete band gaps (CBGs) [3]. Since then, great interest was focused on fabricating threedimensional (3D) photonic crystals (PCs) in order to obtain CBGs. Several methods were reported, and CBGs were achieved in the range of microwave or submicrowave. Theoretical analysis showed that although CBGs can be obtained by diamond structure, a strict condition should be satisfied, i.e., the modulation of the refractive index of the material used should be larger than 2.0 [4,5]. So, attention was paid to finding the materials with high refractive index. Some scientists tried to fill the templates with high refractive index materials to increase the

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viii

Venla E. Laine

modulation of the refractive index [6-8], and CBGs were obtained. However, the CBGs achieved in 3D PCs were mostly in microwave or infrared regions [7,9-11]. Holography is a cheap, rapid, convenient, and effective technique for fabricating 3D structures. In 1997, holographic technique was introduced for fabricating the face centered cubic (fcc) structure [12]. Campbell et al. actually fabricated the fcc structure with holographic lithography [13]. Several authors reported their works on this topic [4,5,9,14]. Toader et al. also showed theoretically a five-beam “umbrella” configuration in the synthesis of a diamond photonic crystal [15]. Because both the value and modulation of the refractive index of the holographic recording materials are commonly low, there would be no CBGs in PCs made directly by holography. For example, the epoxy photoresist generally used for holographic lithography [13,16-18] has n=1.6, which is a little bit too low. They may, however, be used as templates for the production of inverse replica structures by, for example, filling the void with high refractive index and burning out or dissolving the photoresist [13], and a good work was done by Meisel et al [19,20]. However, to find materials with large refractive index is not easy, and the special techniques needed are very complicated and expensive. It limits the applications of PCs, especially for industrial productions. Although some efforts had been made [20], CBGs in the visible range had not yet been achieved by using the materials with low refractive index. Therefore, it is a big challenge to fabricate 3D PCs possessing CBGs in the visible range by using materials with low refractive index, though it is greatly beneficial for the future PC industry. As a first step for achieving this, it is important to obtain very wide band gaps. In the previous investigations [21,22], it was evidently shown that the anisotropy of a photonic band gap in a two-dimensional photonic crystal is dependent on the symmetry of the structure, and as the order of the symmetry increases, it becomes easier to obtain a complete band gap. One would naturally ask whether or not such a method is applicable to 3D PCs. In view of this question, in Chapter 2 the authors proposed a series holographic method for fabricating some special structures by using materials with low refractive index, and the features of the band gap in such structures were then studied experimentally and theoretically. Chapter 3 introduces photonic crystals for microwave, millimeter wave and submillimeter wave applications. First, the authors present a new type of photonic crystal waveguide for millimeter or submillimeter waves which consists of arrayed hollow dielectric rods. The attenuation constant of the waveguide is smaller than that of a Nonradiative Dielectric (NRD) Waveguide. Second, metallic photonic crystals consisting of metamaterials are designed. Metamaterials are artificially-constructed materials whose permittivity and/or permeability can be negative. A hollow metal rod with a slit in its sidewall is a metamaterial which plays the role of an LC resonator. In addition, an array of such metal rods can behave as a twodimensional photonic crystal. Conventional metallic photonic crystals have a photonic bandgap for electromagnetic waves with E-polarization (where the electric field is parallel to the axis of the rods). On the other hand, metamaterials consisting of hollow metal rods with sidewall slits have a bandgap for electromagnetic waves with H-polarization (where the electric field is perpendicular to the axis of the rods) because of their negative permeability. Eigenvalues can be calculated using the two-dimensional finite-difference time-domain (FDTD) method, and metallic metamaterial-based photonic crystals can be designed with independently customized bandgaps for E- and H-polarizations.

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Preface

ix

Third, the authors analyze propagation losses in metallic photonic crystal waveguides. A conventional metallic waveguide structure whose E-planes are replaced by metal wires can easily confine electromagnetic waves because their electric field direction is parallel to the wires. In this section, the attenuation constants of metallic photonic crystal waveguides, whose E-planes are replaced by arrayed metal rods, are calculated and compared with those of conventional metallic waveguides. Losses are calculated using another FDTD method. The propagation losses are found to decrease when the radii of the rods (relative to the lattice constant) is increased. The bandwidth can be extended by widening the waveguide relative to the lattice constant. Fourth, the authors investigate a metallic waveguide in which in-line dielectric rods are inserted. In such a situation, the electromagnetic field is equivalent to that produced by a twodimensional photonic crystal with a lattice constant in one direction equal to the inter-rod spacing, and in the other direction equal to twice the distance between the rods and the sidewalls. The calculations of the fields are performed using the FDTD method. The authors’ results showed that the frequency range of a single propagation mode in the waveguide can be extended. Finally, arrayed dielectric rods in a metallic waveguide are known to change the modes and of course the group velocities of electromagnetic waves. In particular, the authors focus on TE10 to TEn0 (where n equals 2 or 3) mode converters and the dependence of their behavior on the frequency range. Conventional metallic waveguides only allow propagation at frequencies above the cutoff frequency fc. The TE20 mode, however, is also possible for frequencies higher than 2 fc. In this investigation, a mode converter is proposed which passes the TE10 mode for frequencies less than twice the cutoff frequency, and converts the TE10 to the TE20 mode for frequencies higher than twice the cutoff frequency; this is achieved by small variations of the group velocity. In Chapter 4, properties of the directional coupler made by photonic crystals (PCs) are studied by the tight-binding theory (TBT), which considers the coupling between defects of PCs. Based on this theory, the amplitude of the electric field in a photonic crystal waveguide (PCW) can be expressed as an analytic evolution equation. As an identical PCW is inserted with one or several partition rods away, the PC coupler is created. The nearest-neighbor coupling coefficient  between defects of the coupler causes the splitting of dispersion curves, whereas the next-nearest-neighbor coefficient causes a sinusoidal modulation to dispersion curves. The sign of determines the parity of fundamental guided modes, which can be either even or odd, and the inequality |  | < 2 |  | is the criterion for occurring crossed dispersion curves. There is no energy transferred between the PCWs at the frequency of the crossing point, named as the decoupling point, of dispersion curves. By translating the defect rods along the propagation axis of the coupler, blue shift (red shift) in the frequency of the decoupling point occurs to the square (triangular) lattice. Therefore, the frequency of the decoupling point and coupling length of the coupler can be adjusted by moving defect rods. By applying propagating fields having a frequency at the decoupling point and another frequency where the coupler has an ultra short work region, the authors designed a dual-wavelength demultiplexer with a coupling length of only two wavelengths and power distinguish ratio as high as 15 dB. An analytic formula can also be derived by the TBT for asymmetric couplers made of two different coupled PCWs. The asymmetric coupler possesses the following properties: (1)

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Its dispersion curves will not cross at the “decoupling point” and the electric field would only localize in one PCW of the coupler; (2) The eigenfield at the high (low) dispersion curve always mainly localizes in the PCW that possesses high (low) eigenfrequency, even though the symmetry of eigenmodes has changed. As the field with a given frequency is incident into one of the PCWs of the coupler, both the energy transfer between two waveguides and the corresponding coupling length can be given. If an optical Kerr medium is introduced into a symmetric coupler, the coupler becomes an optical switch through the modification of the refractive index of the coupler by sending a high-intensity field. The most interesting phenomena in photonic crystals stipulate the presence of omnidirectional band gap, i.e. overlapping of stop bands in all directions. A higher rotational symmetry and an isotropy of quasicrystals in comparison with ordinary crystals give a hope to achieve a gap opening at lower dielectric contrasts. But nonperiodic nature of quasicrystals makes the size of stop bands lower than in the case of an ordinary periodic crystal. Another possibility to get a higher lattice isotropy (symmetry) is to use the so called quasicrystal approximants and periodic structures with a large unit cell. Within such a unit cell quasicrystal approximants are nearly identical to the actual quasicrystal. Outside of that unit cell, they qualitatively resemble the quasicrystal, but in a periodic manner. Chapter 5 is devoted to investigation of a photonic band structure of such lattices. The transition from periodic structure to nonperiodic one is studied for 2D case by considering quasicrystal approximants of growing period. It is done in order to weigh advantages and disadvantages of quasicrystals. It is shown, that higher isotropy in the 2D case allows one to reduce the refractive index threshold necessary for the gap opening For the 3D case the authors consider different approximants of icosahedral quasicrystals with 8 and 32 "atoms" per unit cell and Si-34 structure with 34 "atoms" per unit cell. All considered structures demonstrate a photonic gap opening and a high isotropy of the band structure. At the same time there are no structures with refractive index threshold for gap opening lower than the best cases of periodic lattices, in particular, with diamond symmetry. In Chapter 6, the effect of different controlling parameters such as number of periods of the structure, refractive index contrast, filling fraction and angle of incidence of light on the structure on the output spectrum of one-dimensional (1D) photonic crystal has been shown and discussed. By knowing the effect of these parameters on the output spectrum, one can engineer the output spectrum of the 1D photonic crystal to utilize them in various optical applications such as filters, reflectors etc., in a desired and optimum way. The transmission properties of guided waves in photonic crystal waveguides containing barriers formed from Kerr nonlinear media are studied theoretically. The photonic crystal waveguides are formed in a two-dimensional (square lattice) photonic crystal of linear dielectric cylinders by cylinder replacement, with replacement cylinders made of linear dielectric media. The channel of the waveguide is along the x-axis of the photonic crystal. Barriers formed of Kerr nonlinear media are introduced into the waveguide by cylinder replacement of waveguide cylinders in the barrier region by cylinders containing Kerr nonlinear media. The transmission of guided modes incident on the Kerr media barriers, from the linear media waveguides, is computed and studied as a function of the parameters characterizing the Kerr nonlinear barrier media. A focus of Chapter 7 is on the excitation of intrinsic localized modes within the barrier media and on the effects on these modes from additional off-channel and in-channel features that are coupled to the barrier media. Systems treated are a simple Kerr nonlinear media barrier formed by replacing waveguide sites by

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Preface

xi

cylinders containing Kerr nonlinear media, Kerr barriers that couple to off-channel impurity features formed by cylinder replacement of off-channel sites of the photonic crystal which are adjacent to the barrier, and Kerr barriers containing an in-channel impurity site. The offchannel and in-channel impurity features may be formed of linear media and/or Kerr media replacement cylinders and are found to give rise to multiple bands of intrinsic localized modes. Suggestions are made for possible technological applications of these types of systems. Colloidal dispersions with self-assembling properties have been prepared by soap-free emulsion copolymerization of styrene (St) with hydroxyethylmethacrylate (HEMA), using various St/HEMA molar ratios. Investigations by SEM, AFM and reflexion spectra have shown that copolymer particles give colloidal crystals while monodisperse particles of polystyrene, with no HEMA units, do not crystallize. XPS analysis of the copolymer particles has shown that the HEMA hydrophilic units tend to concentrate on the surface of the copolymer particles, while the core is richer in St. The copolymer particles have been separated and redispersed in other liquids than water (ethylene glycol, formamide, ethanol, hexane, and toluene) to investigate the conditions that allow self-assembling after the diluent evaporation. For each case, contact angles have been measured and the adhesion work has been computed. The results in Chapter 8 have shown that the contact angle does not influence the self-assembling process, while the adhesion work is an important parameter. Only dispersions characterized by a value of the adhesion work superior to 87 mJ/m2 did crystallize after the diluent evaporation. Taking into account that the current development of the world requires Increasingly devices operating at higher frequencies and bandwidths, and that these requirements are no longer possible to be obtained by electronic, in Chapter 9 the authors propose and analyze an optical logic gate based on nonlinear photonic crystal (PhC). This device uses an optical directional coupler. In the authors’ simulations the authors used the following methods: PWE (Plane Wave Expansion), FDTD (Finite-Difference Time-Domain), and COMSOL, which is integrated to their own BiPM (Binary Propagation Method). During the past two decades, there has been considerable attention paid to materials with periodic modulation of the refractive index. Such structures, called photonic crystals, can have periodicity in one, two or three dimensions and typically utilize a period that is approximately one half of the operating wavelength. One dimensional photonic crystals have been used in practical devices inlcuding distributed feedback lasers [28, 50] and vertical cavity surface emitting lasers [64] which are utilized in applications ranging from optical communication systems to laser printing. Devices based on two and three dimensional photonic crystals remain an ongoing research topic. Chapter 10 discusses the use of two dimensional photonic crystals for integrated photonics applications. In particular the authors focus on the band structure and modal properties of waveguides and cavities that posess nonsymmorphic space group symmetry in the form of a glide plane. In Chapter 11, structural, optical and nonlinear optical properties of strontium tetraborate (SBO) single crystals are summarized. SBO presently is not considered to be a ferroelectric, however, domain structures consisting of alternating oppositely poled domains, strongly resembling those present in such ferroelectrics as potassium titanyl phosphate and lithium niobate, were recently discovered in this crystal. Such geometrical properties as orientation, size, and degree of randomization, of these domain structures are described. The problem of origin of domain structures in SBO is considered, as well as the possibility of their

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characteristics control. From the point of view of optical properties, domain-structured samples of strontium tetraborate are classified as randomized nonlinear photonic crystals. Comparative study of nonlinear diffraction and random quasi-phase-matching of nanosecond and femtosecond laser pulses in these nonlinear photonic crystals is presented. Prospects of creation of nonlinear optical converters of laser radiation into VUV spectral region based on domain structured strontium tetraborate are discussed. Photonic crystals are composite materials possessing a periodical modulation of their dielectric constant on the scale length of the visible wavelength. This dielectric periodicity affects light propagation creating allowed and forbidden photon energy regions. Among different types of photonic crystals, the authors addressed their attention to artificial opals i.e. three dimensional self-assembled arrays of spheres packed in a face centred cube crystal lattice. In order to tune the optical properties of such photonic crystals, the authors inserted gold nanoparticles in the interstices between the spheres. The authors call this process doping. In Chapter 12 the authors report on the optical properties of artificial opals doped at various levels with gold nanoparticles (NpAu) stabilized with citrate molecules. By increasing the NpAu doping level, the opal photonic stop band was bathochromically shifted as well as its full width half maximum and intensity were reduced. No effects of nanoparticle coalescence was observed as previously detected for NpAu prepared by laser ablation. In order to improve the mechanical stability of NpAu doped opals, the authors started a preliminary thermal annealing study with in situ monitoring of optical properties. The authors found that thermal degradation of doped samples was reduced with respect to bare opals. In addition, the annealing temperature of NpAu doped opals was much higher than that observed for bare ones. In the Short Communication, the authors analyze the mathematical structure of the refractive group index (briefly, group index) of a photonic crystal starting from the fact that, in a photonic crystal, the index of refraction as a function of wavelength is a periodic function whose period is the atomic lattice spacing of the crystal so that the expansion in a Fourier series of the refractive-index function is considered. In particular, the case in which the wavelength is sufficiently small (geometrical optics) is examined.

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In: Photonic Crystals Editor: Venla E. Laine, pp. 1-29

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Chapter 1

FABRICATION AND APPLICATIONS OF POLYMER PHOTONIC CRYSTALS Jingxia Wang and Yanlin Song* The Laboratory of New Materials, Key Laboratory of Organic Solids, Institute of Chemistry, Chinese Academy of Sciences

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1. Introduction Photonic crystals (PCs) are defined as a periodic dielectric structure with a refraction index varying on the order of the wavelength of light, which was firstly described by Yablonovitch[1] and John[2] in 1987. Due to the periodic variation of refractive index, the crystals possess unusual optical properties------photonic bandgap (PBG), a range of frequencies, in which light (with certain wavelength) is forbidden to exist within the bulk of the PCs, and light with other wavelength may propagate through PCs uninterruptedly. PCs are also called light-semiconductors due to its optical analogue of the electronic bandgap in semiconductors. The function of special light manipulation endows PCs many potential applications. According to their periodicity in space, PCs can be categorized mainly into onedimensional (1D), 2D and 3D PCs. All above-mentioned PCs have been fabricated artificially from a variety of materials. To date, the main 3D PCs include opal, inverse opal, woodpile, diamond structure, etc. Among those structures, opal structure comprised of polymer colloidal crystals demonstrates promising advantages over other structures, for example, polymer colloidal is easily processed, and its properties could be easily modulated based on the design of molecular structure. Moreover, polymer colloidal crystal can also act as template for the fabrication of PCs of inorganic, metal and complicate materials, since nanometer sized polymer particles are readily available with a high mono-dispersity or can be synthesized in a controlled way. Plenty of review papers[3-9] have been published about the fabrication of polymer PCs and its special applications. Such as, fabrication of polymer PCs form latex system[3-5] inverse opal structure[6], and micro-phase separation[10]. Polymer PC material *

E-mail address: [email protected]. (Corresponding author)

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is a novel material, and can provide material basis for the development of relative subjects, which covers photo-computing, photo-communication and optic-electric integration. In this paper, we review the recent research progress of polymer PCs, mainly concern on the fabrication and application of polymer PCs.

2. Fabrication of Polymer PCs There are generally two approaches for the fabrication of polymer PCs: top-down lithography process[11-12] and bottom-up self assembly approach[13-16]. The former includes mechanical -manufacture, micro-, photo-, electric- lithography and holography etc. This approach is easy for the manufacture of PCs with stopband in the ranges of microwave, but it is expensive and time-consuming for the PCs with stopband at visible or near infrared ranges. Thus, polymer PCs are preferably to be fabricated from self-assembly of latex spheres and its inverse opal or microphase separation of block copolymers (BCPs) for the process is simple, low-cost, and easy duplication. Self-assembly approaches that have been applied to the fabrication of 3D PBG crystals include the following: 1) crystallization of spherical or non-spherical building blocks such as monodisperse colloids; 2) template-directed synthesis against a close packed lattice of spherical colloids, the structure of the resultant PCs depend on the PCs template; 3) Phase separation of BCPs. In this section, several typical fabrications of polymer PCs were presented based on various fabrication principles.

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2.1. Self-Assembly of Latex Spheres from Gravity Polymer PCs, in the earliest report, were self-assembled by natural deposition under gravity field[14]. This process takes longer time (several weeks or months), and is cannot control the assembly process, so that the resultant sample consists of mixing crystal structure. Later, the PCs were obtained via centrifugation or filtration under high vacuum, which could greatly accelerate the latex nucleation and transfer processes.[15] To further control the fabrication process, Nagayama et al. firstly put forward the fabrication of polymer PCs from convective assembly, and investigated in detail various inter-particle forces governing the packing of two-dimensional latex aggregates,[17-20] and found that the water evaporation and transfer convective flow played a crucial role on the latex arrangement. This method could effectively control crystal procedure by evaporation rate. Subsequently, Jiang[21] further simplified the fabrication process: vertically putting a glass substrate into the latex suspension, the latex sphere will be well-ordered transferred upon the substrate during evaporation procedure, the so-called vertical deposition method greatly boosted the wide application of this self-assembly approach. In this process, the crystal thickness could be well controlled by the concentration of the latex suspension or latex diameter. To meet application requirement of high quality PCs, many researchers took much efforts to further optimize the crystal procedure. Ozin et al. introduced the mechanical stirrer[22] or temperature gradient[23] into the constant temperature deposition system.[24] These methods are suitable for the assembly of latex spheres with diameter of larger than 800 nm.[22-25] Gu[26] achieved a fine control of the crystal thickness by using lifting apparatus, the apparatus could effectively control the thickness by modifying the lifting rate or latex concentration.

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Fabrication and Applications of Polymer Photonic Crystals

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Fudozui[27] removed the uneven crystal structure by covering the hydrophobic liquid upon the contact line of the substrate. Meng[28] put forward capillary assembly of latex spheres and ultra-fine particles (Figure 1). Recently, they also demonstrated a fabrication of PCs with high quality by pressure controlled isothermal heating vertical deposition method[29], and the films showed deep photonic band gap and steep photonic band edge.

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2.2. Self-Assembly of Latex Sphere by Exterior Field Various exterior fields, such as electric [30-35] or magnetic field, are also used for selfassembly of polymer PCs with special properties. There are mainly three kinds of electric action for the fabrication of colloidal PCs, such as, electrostatic[30], electrophoresis[31-33] or electrochemical[34-35]. Highly charged monodisperse latex particles could form well-ordered 3D arrangement by electrostatic interaction [30] and lateral capillary force. The obtained crystalline structure could be further modified by ion or electrolytes concentration. This slow assembly process required stringent assembly condition. Latex assembly via electryphoresis[31-33] is a rapid and controllable assembly way for the charged latex spheres. The assembly process could be well controlled by electronic strength and viscosity of liquid media. While electrochemical assembly [34-35] method has special advantages, such as rapid assembly and controllable process. This method is especially suitable for complex substrate, and prefers to the semiconductor materials with high refractive index (CdS, CdSe, II-IV, IIIV, add IV). The resultant films show high mechanical strength and good heat-resistance. The earliest fabrication of PCs based on electric is reported by Yeh et al.[31] They assembled ordered colloidal aggregates by electric-field-induced fluid flow, in which, the highly charged monodisperse latex particles were applied among two electrodes. The polarization of the bead double layer in the external field produce the repulsive interaction between beads, and electroosmetric flow yields attractive interaction among beads. Velev et al.[33] developed colloidal crystal from alternating electric fields by two coplanar electrodes (Figure 2). This assembly is driven by forces resulted from the electric field gradient. The assembly process can be controlled via the field strength, frequency, and viscosity of the liquid media.

Figure 1. Schematic illustration of capillary assembly of latex spheres and ultra-fine particle. [28] Copyright 2002 American Chemical Society

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Jingxia Wang and Yanlin Song

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Figure 2. (a) Schematics of the experimental arrangement and dimensions of the experimental cell. (b, c) Schematics of the two-stage mechanism of crystallization deduced from the set of diffraction and microscopy data. (b) Immediately after the field is applied, the particles align in chains due to the induced dipole-dipole interactions. Simultaneously, the gradient-driven dipole-field force pulls the chains and particles toward the surface of the substrate between the electrodes. (c) The particle chains confined on the surface crystallize to form 2D hexagonal crystals aligned by the field. [33] Copyright 2004 American Chemical Society

In addition, self-assembly of latex spheres by magnetic field is another effective approach for the fabrication of PCs with special properties. Typically, Bibette et al. reported that emulsion droplets containing γ-Fe2O3 nano-particles can form one dimensional chain structure and diffract light in the visible range in the presence of an external magnetic field.[36-37] Asher et al. have developed more stable building blocks by embedding superparamagnetic iron oxide nanoparticles into monodisperse polymer colloids through an emulsion polymerization process.[38-39] While Yin et al.[40-43] applied pure magnetic materials as building blocks of colloidal crystal, which is favorable to a faster response to the external field. They synthesized a polyacrylate-capped superparamagentic colloidal nanocrystal clusters of magnetite (Fe3O4) with tunable size from 30 to 180 nm. The Fe3O4 colloidal nano-crystal clusters readily self-assembled in deionized water upon application of external magnetic field[40] , and diffract brilliant visible light when the field strength is changed. Further modification of colloidal nanocrystal cluster particles with a layer of silica allows their dispersion in various polar organic solvents[41] such as methanol, ethanol and ethylene glycol. The modified particles can still self-assemble into ordered structure in these nonaqueous solvents[42] and diffract light upon application of an external magnetic filed.

2.3. Self-Assembly of Latex Spheres Under Physical Confinement [44-52] The crystal structure fabricated from above-mentioned method is face centered cubic structure. Its stopband is narrow, and the displacement of the crystals structure may close the stopband. Self-assembly of the latex spheres based on template effect [44-47] make it feasible for controllable crystal orientation. Blaaderen[44] fabricated PCs film with uniform thickness

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by combining microcavity template and evaporation effect. Amos[45] obtained large-area single crystal by applying controlled shear upon the substrate. Cheng[46] applied the template gradients to control the nucleation and growth of hard–sphere colloidal crystals (CCs). Velikov[48] obtained binary colloidal crystals (CCs) with control over the crystal orientation through a simple layer-by-layer process. Well-ordered single binary colloidal crystals with a stoichiometry of large (L) and small (S) particles of LS2 and LS were generated. The formed structure is mainly based on the templating effect of the first layer and the forces exerted by the surface tension of the drying liquid. Different from above method, Xia[49] designed a special apparatus ( Figure 3) for the fabrication of crystalline lattices of mesoscale particles over ~1 cm2. The apparatus could effectively control the crystal structure. The apparatus includes the sample cell fabricated by sandwiching a photoresist frame between two glass substrates. A tube injecting latex suspension was connected with the upper glass, while water leakage tube was connected to the bottom glass. Photolithography films provide the suitable channel for the latex assembly, while the exterior pressure through injection entrance of upper glass could accelerate latex assembly and water leakage, the additional ultrasonic could induce latex arrangement. This apparatus combines the complicate effects of the filtration, ultrasonic, shake and template, and will benefit for the well-ordered assembly of the latex spheres. Based on this work, Xia[50-51] developed series of complex aggregates of monodisperse colloids with welldefined sizes, shapes and structure by taking advantage of various physical templates, such as, cylindrical hole, square, rectangular, trenches, triangular and noncircular pattern. Ozin[52] also obtained various colloidal crystal by using the predefined silicon template and centrifugation action.

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2.4. PCs Fabricated from Micro-Phase Separation Different from above-mentioned self-assembly approach, fabrication of PCs from microphase separation needs BCPs rather than monodisperse latex particles. BCPs are interesting platform materials for PC applications due to their ability to self-assemble into 1D, 2D and 3D periodic structures and the ability to selectively incorporate additives. The simplest example for this kind of PCs is the multilayer stack, whose wavelength mainly depends on the optical thickness of each layer. BCPs with compositions of 35-65% can selfassemble into a periodic lamellar structure. The period scales are of two-thirds power of the molecular weight. Alternatively, when PCs are made from phase separation of homopolymerBCP blends, its length scales can be increased via swelling of homopolymer, its stopband can be conveniently tuned in an approximately linear manner by simply adjusting the amount of homopolymer.[53-54] Varying the refractive index of the layers can modify the band structure, and increasing the dielectric contrast will lead to larger bandgaps. Self-assembly from a pair of immiscible homopolymers and relatively high molecular weight BCPs has produced 2D[55] cylindrical domains and 3D-body-centered cubic packed spherical domains. Thomas et al. have prepared three known types of micro-domain structures with cubic symmetry: spheres, double gyroid and double diamond[56-57]. The double gyroid microdomain structure is comprised of two interpenetrating, 3-coordinated labyrinthine networks of the minority lamellar and cylindrical phases for BCPs. Figure 4 exhibits the periodic structure obtained from microphase separation of BCPs of Polystyrene (PS)-b-polyisoprene (PI))[57-

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58]， the periodic structure of PS and air could be obtained when removing PI by UV irradiation. The complex double gyroid structure could be fabricated by applying triblock polymer (PS-PI-polymethyl methacrylate(PMMA))[59]. The polymer materials not only displays anisotropic dielectric properties but also is able to vary the anisotropy by applied fields, which points the way towards the use of self-assembled block polymeric materials in optical switches, couplers and isolators.[60] There are four main challenges that need to be overcome in order to achieve necessary photonic properties with BCPs. The primary challenge is to obtain the large domain sizes needed for the optical frequencies of interest. Additional significant concerns for creating useful BCP PCs include long-range domain order and domain orientation, as well as sufficient dielectric contrast between domains.

Figure 3. Schematic illustration of experimental set-up for physical confinement approach. [49] Copyright wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

Figure 4. [5] Schematic illustrations of some typical mesoscale periodic structures that could be generated through the phase separation of organic BCPs (in this case, PS±PI). By varying the ratio between the chain lengths (or molecular weights) of these two units, one could obtain long-range ordered phases such as the 3D body-centered cubic lattice of PS spheres; the 2D hexagonal array of PS cylinders; the three-dimensionally ordered bicontinuous double diamond structure; and the 1D lamellae. [5] Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

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2.5. Large-Scale Fabrication of Polymer PCs Although above methods put forward some effective fabrication of polymer PCs with high quality, it is still difficult for the large-scale fabrication of PCs relating to real practical application. For example, it will take long time for large-scale samples, while the long-time soaking in the latex suspension would rust metal substrates, e.g., iron substrate. Regarding of these problems, Jiang et al. firstly put forward the fabrication of polymer PCs by spin coating, which is easy for the fabrication of PCs with centimeter-size within minutes.[61] The largescale sample could be used as template for the inverse opal, and this method is also be used for the fabrication of PCs with special morphology[62]. However, this method needs highly viscous latex suspension, where, the re-dispersion of latex spheres is tedious,[62] and the application of special dispersant would impair optic properties of the resulted films due to the lower refractive-index contrast.[61] Moreover, the size of the resultant films is circumscribed to the area of the top surface of the spin coating instrument. Song et al.[63] recently developed an ultrafast fabrication of large-scale polymer PCs by spray coating (Figure 5). They realized a rapid assembly of the latex sphere during short spray process by taking advantage of latex spheres with hydrophobic PS core and hydrophilic PMMA/polyacrylic acid(PAA) shell, and the latex shell with abundant COOH groups resulted in strong hydrogen bonding interaction among latex spheres, which boosted latex arrangement during the ultrafast spray procedure. As a result, the large-area colloidal PCs could be rapidly fabricated on different substrates by spray coating. The resultant colloidal PCs showed well-ordered latex arrangement and iridescent color. This ultra-fast fabrication procedure will be of great importance for practical application of PCs in the fields of optic devices and functional coatings.

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2.6. Fabrication of Patterned Polymer PCs by Printing Patterned PCs[41, 43, 52, 64-65] have attracted great attention because of their broad applications in optical devices,[66-72] displays,[73-76] and microfluidic devices.[77] Generally, the patterned PCs can be achieved by lithographic approaches[78-79] and selfassembly methods.[80-81] But the traditional lithographic approach was complex, timeconsuming and expensive, while the patterns of the PCs fabricated from the self-assembly method, were limited by the pattern of the template.[82] Ink-jet printing, a particularly attractive patterning technique for the manufacture of high performance devices,[83-87] has been utilized for the fabrication of pattern PCs by Moon and Song et al. Moon fabricated PC micro-arrays composed of two different-sized latex spheres by using single-orifice ink-jet printing.[88-92] This facile fabrication of patterned PCs by combining the self-assembly and direct write was regarded to be promising for the realization of nano/microperiodic structures for next generation photonic and display devices. To further improve the optic properties of the patterned PCs and accelerate the application of this method. Song et al.[93] fabricated large-scale patterned PCs from common ink-jet printer(Figure 6). In this work, the latex suspension with special latex structure was used as ink, and the printing substrates with suitable wettability were applied. The pattern designed from computer software could be directly printed. These work demonstrated the first fabrication of multi-stopbands macroscale patterned PCs from common ink-jet printing. This rapid fabrication of multi-stopbands PCs

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will be of great significance for extensive applications of PCs. Just recently, Kim and Yin[9495] fabricated a high-resolution patterning of multiple structural colors within seconds, based on successive tuning and fixing of colour using a single material along with a maskless lithography system. The colour of films is tunable by magnetically changing the periodicity of the nanostructure, and the pattern structure of the films is photochemically immobilized in a polymer network. This simple, controllable and scalable structural colour printing may have a significant impact on colour production for general consumer goods.

3. Polymer PCs with Special Properties

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The periodic dielectric structure of polymer PCs endows the films special light manipulation properties, which have potential applications in various optic devices. While application properties of films can be effectively modulated by applying functional polymers. In this section, we mainly refer to some research work about the films’ mechanical strength, wettability, and its optic properties.

Figure 5. The schematics of fabrication procedure for colloidal PCs by spray coating method (the insert indicated the typical hydrogen bonding interaction resulting from latex spheres with abundant COOH groups). [63] Copyright Wiley-VCH Verlag GmbH & Co. KGaA. produced with permission.

Figure 6. (a) Photo of the PCs with flower-leaf pattern, the scale bar: 1 cm, (b) SEM images of greenleaf (220 nm) pattern region (scale bar: 1 um). [93] Reproduced by permission of The Royal Society of Chemistry.

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Figure 7. Tough PCs made from latex spheres with hard PS core-flexible PMMA/PAA shell. [102] Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

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3.1. Polymer Pcs with High Strength Generally, polymer PC films suffer poor mechanical strength[24, 96] due to the interstice structure among latex spheres. In order to improve the applicability of PCs under various circumstances, many measures have been taken to enhance the linkage among latex spheres[3, 97-105]. For instance, Xia[97] and Blanco[98] sintered the PC films at high temperature to decrease pore size. Ozin[99] allowed the growth of a continuous silica layer covering the film by chemical vapor deposition. Ruhl[100] and Wang[101] fabricated a tough colloidal PCs with hard PS core structure and soft PMMA/PAA shell (Figure 7), in which, the hard PS core keep well-ordered arrangement, and the adhesion effect of soft and flexible PMMA/PAA shell will enhance the linkage of latex spheres. While the tough closed-cell polyimide inverse opal structure[102] could be fabricated when applying this core-shell PSPMMA-PAA latex spheres aggregates as PC template. The obtained polyimide PCs with a closed-cell structure had much better mechanical properties than those with open-cell structures. Some literatures[104-105] used flexible (poly(N-isopropylacrylamide)(PNIPAM) as latex shell to improve the mechanical properties of the CCs. Wu et al[106] fabricated flexible and tough PC films from building blocks of a soft polymer core and hard SiO2 nanoparticle shell. All these methods produced great enhancement for the mechanical properties of the PC films. However, the resultant films show poor solvent-resistance due to the intrinsic properties of polymer materials. This greatly restricts their practical application, especially in some solvent systems where PC films were required. Inspired from weather-resistance properties of natural PCs, such as feather of peacock, wings of butterfly, whose tough properties mainly be derived from their crosslinked network structure, Song et al.[107] developed a common method to improve both the mechanical strength and solvent-resistance properties of PCs by photocrosslinked approach. The photopolymerized monomers were infiltrated into the interstice of

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latex spheres and subsequent photo-polymerization after UV irradiation. The obtained elastic and flexible crosslinked polymer network could enhance the linkage among latex spheres, which greatly improve both the mechanical strength and solvent resistance of the resultant PCs. Furthermore, this simple photo-crosslinkage enhancement approach can be effectively extended to most PCs, such as, PS and SiO2 PCs. This approach opens a facile way for improvement of both mechanical strength and solvent resistance of PCs, and shows promising implication for the practical application of PCs.

3.2. Polymer PCs with Special Wettability[108]

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There are many PC structures in nature, such as natural opal, hair of seamouse, feather of peacock and wings of butterfly. These PCs have not only structure color from their periodic structure, but show special wettability. Taking a butterfly wing as a example, its wings show self-cleaning properties as lotus, and also the wings exhibit iridescent color upon sun’s irradiation. Inspired from these natural phenomena, Gu et al.[109-111] fabricated PCs with both structural color and self-cleaning properties based on uniform inverse opal structure over a large area. In this case, the structure color is derived from their periodic micro-porous structure, while the special wettability results from the combination of both hydrophobicity/hydrophilicity surface and roughness of periodic structure.

Figure 8. (a) AFM image of as-prepared superhydrophobic inverse opal film, (b) water droplet profile on as-prepared film (a) [109] Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission. (c) SEM image of as-prepared superhydrophilic film, (d) the change of water CA with storing time, dashed lie is films without pore, while solid line is the film of CCs. Reprinted with permission from [111]. Copyright [2004], American Institute of Physics.

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The superhydrophobic film[109] was obtained when coating the inverse opal structure with fluoroalkylsilane by thermal chemical vapor deposition or directly applying of hydrophobic PS PCs[112]. The as-prepared inverse structure demonstrated a hexagonal arrangement of the air spheres encircled by the silica networks (Figure 8a). Such 3-D ordered structure contributes to the distinct stopband, and the roughness surface from periodic structure amplifies the hydrophobicity of fluoroalkylsilane surface, all of these result in a colorful superhydrophobic film shown in Figure 8b. While superhydrophilic surface was obtained based on hydrophilic TiO2 inverse opal[111] in Figure 8c and 8d. Furthermore, the wettability of the films could be controlled by varying the roughness structure based on the changes of latex spheres and its spacing. For example, binary colloidal assembly[113-114] could enhance the roughness of the films surface, which will be favorable to the formation of superhydrophobic surface. Shiu et al.[115] achieved wettability modulation of films surface by gradually decreasing the latex diameter of close-packed PS nanostructure by oxygen etching, which could effectively modify liquid-solid contact area fractions. PCs with responsive wettability could be obtained when PCs were made of responsive materials or its surface was coated with responsive polymer. In this case, the change of the surface chemical composition arisen from stimuli leads to the change of the wettability, while roughness structure will amplify the wettability change of the films. For example, Wang et al.[116-117] fabricated a PCs with tunable wettability by using latex spheres with amphiphilic materials of PS-b-PMMA-b-PAA. The wettability of the films can be tuned from superhydrophilic to superhydrophobic by varying the assembly temperature. The change of wettability could be mainly attributed to the change of the surface chemical composition driven from phase separation of polymer segments toward minimum interfacial energy when increasing assembly temperature. This phase reversion procedure results in a gradual increase of the water contact angle as the assembly temperature rises. In addition, the change trend of wettability for the PCs can be finely controlled by adjusting the ratio of the soft polymer segment and hard polymer segment,[117] which can modify the phase change temperature of the polymer. Otherwise, the wettability of the film could be flexibly controlled via adjusting the pH of the assembly system[118], in which the variation of pH modified the presence or not of the hydrogen bonding between hydrophilic groups (Figure 9). The stable superhydrophobicity arises from hydrogen bonding association between SO3-Na+ of sodium dodecylbenzene sulfonate (NaDBS) and hydrophilic COOH around the latex surface, and the association locks hydrophilic groups of the latex sphere surface into a preferable configuration. Vice versa, the superhydrophilic films were achieved when introducing NH3H2O into the latex suspension (pH=12), in this case, hydrogen bonding is suppressed due to deprotonation of COOH to COO-. Similarly, PCs with photo-responsive wettability have been obtained by Ge et al.[119] which were achieved by electrostatic layer-by-layer selfassembly of photoresponsive azobenzene monolayer upon the silica inverse opal. And polypyrrole[120] inverse opal PCs with electrochemical response could be obtained, and its wettability could be electrically tuned from 48.7° to 139.4° when being altered between neutral and oxidation state. In this case, the films show hydrophilicity at the neutral state due to the introduction of hydrophilic Li+, while the films show hydrophobicity at the oxidation state due to the doping of DBS- with a hydrophobic long alkyl chain. The typical application of polymer PCs with special wettability is visibly liquid manipulation properties[121] and patterned fabrication.

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3.3. Polymer PCs with Stopband Modification PCs have aroused wide research interest due to their important application of their optic properties. Particularly, the stopband of PCs could be reversibly changed in response to exterior stimulus. The responsive PCs were generally obtained by introducing various functional materials into PCs system, thus, the change of functional material responsive to exterior stimuli can result in the variation of the refractive index or crystal structure, which will lead to the modulation of stopband. The change of stopband of PCs mainly depends on the Bragg law[122]:

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mλ = 2ndsinθ

(1)

Figure 9. (a and b) Photographs of water droplet shape on the films assembled from suspensions with pH of 6.0 and 12, respectively; (c and d) illustrations of the structure of the latex sphere in the films of parts a and b, respectively. The conformation of hydrogen bonding is noted by the arrow. [118] Copyright 2006 American Chemical Society.

Here, θ is the angle of incidence between the beam and the lattice plane, λ is the wavelength of the reflectance peak maximum (i.e., the stopband position), m is the order of the Bragg diffraction (i.e., m=1), d is the interplanar spacing between (hkl) planes (i.e. the 111 planes), n is the average refractive index of the photonic structure. Therefore, the change of stopband could be attributed to the change of refractive index, interplanar spacing or crystal structure. Bragg diffraction was firstly described for X-rays where atomic lattices caused these interference phenomena.[123] Asher[124-136] firstly fabricated the responsive PCs by coupling PCs with various functional hydrogel materials, in which, the stopband could be modulated based on the volume phase transition of hydrogel induced by exterior stimuli such as temperature,[124] ionic strength,[129] glucose,[133] light irradiation,[126] magnetic field,[38-39] chemical,[132] etc. Some responsive PCs have been fabricated by other group.[137-139] In this process, the lattice space and refractive index can be effectively modulated. For example, photo-responsive PCs could be obtained when introducing azo-benzene into the polymer

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structure. The stopband could be reversibly modified by alternatively irradiated by visible and ultraviolet light.[126] While temperature-sensitive PCs could be obtained by introducing PNIPAM into the polymer PCs, the phase change of the PNIPAM molecular will induce the change of the optic properties of the PCs[124] (Figure 10). The colloidal crystal array (CCA) are highly swollen at low temperature,, almost touching, and diffracting weakly, while the particles become compact and diffract nearly all incident light at the Bragg wavelength above the phase-transition temperature. It is found that the temperature change does not affect the lattice spacing, the 1-nm shift of the maximum wavelength of diffraction upon heating from 10 to 40°C results almost entirely form the change in the refractive index of water. The wavelength-tunable diffraction devices could be obtained based on these volume phase transition properties.

Figure 10. (a) Diffraction from a CCA of PNIPAM spheres at 10 and at 40 °C, insert is temperature switching between a swollen sphere array below the phase –transition temperature and an identical compact sphere array above the transition. (b) temperature tuning of Bragg diffraction from a 125 um thick films　 of　 99　 nm PS spheres embedded in a PNIPAM gel. Reproduced from permission of [124].

On the basis of Asher’s work, Takeoka[140-142] introduced 2 or 3 kinds of functional materials into the same system and the films could demonstrate multi-stimuli response. For example, the thermo-photo responsive PCs have been fabricated from the PNIPAM hydrogel and exhibit a change in hydrophilicity in response to temperature, whereas the minor monomer, 4-acryloylaminoazobenzene, undergoes a change in the dipole moment upon the photo-isomerization of the azobenzene group.[142] Similarly, the PCs can exhibit various switchable colors over the visible region initiated by both electrochemical reaction and temperature change.[140] The films were fabricated from copolymer of electrolyte hydrogel of methacrylic acid (as a pH-sensitive monomer) and NIPAM (as a thermosensitive monomer). This gel reveals an electrochemically triggered rapid two-state switching between two arbitrary structural colors at constant temperature, and its color could be varied based on the temperature change. The color change with electrochemical response mainly can be attributed that electrolysis of water causes a change in pH of an electrolyte solution in the vicinity of the electrodes, the swelling degree of a pH-sensitive hydrogel on the electrodes

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can be easily and drastically changed by electrolysis. The responsive PCs could also be obtained when introducing responsive materials into the porous PC structure. For example, Gu and co-workers fabricated the tunable PCs based on filling organic molecules into PCs, whose reflective index can vary with exterior stimuli such as temperature and light irradiation.[143] The stop band of PCs can also be tuned by pH[144-145] or by magnetic field.[146-147]

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Figure 11.Schematic diagram of the structure of photonic gel film and the tuning mechanism. The photonic gel film was prepared by self-assembly of a diblock copolymer (PS-b-P2VP). Swelling/deswelling of the P2VP gel layers (pink) by aqueous solvents modulates both the domain spacing and the refractive-index contrast, and accordingly shifts the wavelengths of light (hν) reflected by the stop band.

Responsive PCs could be fabricated from microphase separation of functional BCPs. For example, Thomas has fabricated a highly tunable, structural-color reflector based on a hydrophilic/hydrophobic BCP (PS-b-poly(2-vinylpyridine)(P2VP)) that achieved substantial tenability (over 575%) of the primary stop band from ultraviolet (350 nm) to near infrared wavelengths (1600 nm) by changing the salt concentration in the solvent (Figure 11).[131] Recently, they created a full-color pixel by employing the same block copolymer combination with a simple electrochemical cell, the films produce red, green, and blue color through electrochemical stimulation.[148]

4. Applications of Polymer PCs Polymer PCs showed potential application in high performance optic devices and intelligent optic sensing devices based on its special light manipulation properties. For example, the film could be used as colorful coatings based on its structure color, and specific optic device based on its special light manipulation properties. In the following section, some typical application examples of the polymer PCs will be demonstrated.

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Figure 12. Photographs of as-prepared PC hydrogel responding to relative humidity of 20%, 50%, 70%, 80%, 90% and 100%, respectively. [65] Reproduced by the permission of The Royal Society of Chemistry.

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4.1. Optical Sensing Device The basic principle of PC sensor mainly depends on the change of stopband of PCs responding to exterior stimulus. If the stopband of PCs is in the visible range, the response of the films to environmental change could be observed from visible color change. The PCs with special optical properties could provide a new possibility for the chemical and biological detectors. Asher firstly developed visibly PCs sensing device, the color of the films could be reversibly changed in response to light, temperature, pH, ionic species, mechanical, glucose and solvent when coupling PCs with various functional hydrogel materials. Song et al. developed a series of PC sensors[65, 71, 149-150] which could visibly detect environment change, such as humidity[65] or oil kinds,[71, 149] etc. Firstly, they fabricated a colorful humidity sensitive PC hydrogel by infiltrating acrylamide (AAm) solution into poly(StMMA-AA) PC template and subsequent photo-polymerization. As-prepared PAAm -poly(StMMA-AA) PC hydrogel successfully combined the humidity sensitivity of PAAm and structure color of the PC template, it could reversibly vary from transparent to violet, blue, cyan, green and red under various humidity conditions (Figure 12), the color change covering the whole visible range. Otherwise, they fabricated oil-sensitive PCs[149] materials based on phenolic resin (PR) inverse opals with both superoleophilicity and superhydrophobicity due to oil adsorption of PR. The macroporous structure of the inverse opal benefited oil adsorption, and optical properties of PCs could provide various optical signal when adsorbing different oils. As a result, a refractive index variation of 0.02 resulted in a shift of stopband of 26 nm, which showed excellent selectivity for different oils. The visible oil monitor [71] could be realized when selecting the PC materials with higher refractive index, such as carbon materials. Ultrasensive biodetector have been developed based on effective light-manipulation properties. Asher[132] firstly fabricated an interpenetrated CCA glucose sensor by attaching the enzyme glucose oxidase to a Poly colloidal crystal array (PCCA) of PS colloids. Glucose solutions prepared in air cause the interpenetrated CCA to swell and red-shift the diffraction.

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This interpenetrated CCA returns to its original diffraction wavelength after removal from glucose. Asher[151] fabricated PC hydrogel with two coupled recognition modules, a creatinine deiminase (CD) enzyme and a 2-nitrophenol titrating group. Creatinine within the gel is rapidly hydrolyzed by the CD enzyme in a reaction which releases OH-. This elevates the steady-state pH within the hydrogel as compared to the exterior solution. The increased solubility of the phenolate species as compared to that of the neutral phenols causes a hydrogel swelling, which red-shifts the IPCCA diffraction. This PC IPCCA senses physiologically relevant creatinine levels, with a detection limit of 6 íM, at physiological pH and salinity. This sensor also determines physiological levels of creatinine in human blood serum samples. The sensing technology platform may be used to fabricate PC sensors for any species for which there exists an enzyme which can catalyze it to release H+ or OH-. Song et al.[152] verified an ultrasensitive DNA detection using PCs combining fluorescence resonance energy transfer technique (Figure 13). The introduction of PCs into DNA detector system greatly amplified the optical signal; the stopband of PCs inhibited the energy loss of donor molecule. The method can achieve ultrasensitivity up to about 13.5 fm with optimized PCs. Furthermore, the PC-assisted detection method offers a great advantage over conventional techniques for its excellent ability to discriminate single base-pair mismatches, which is of crucial importance for the diagnosis of genetically encoded diseases.

Figure 13. a) DNA sequence detection based on a FRET mechanism. b) Effect of the PC on FRET. c) Emission spectra of dsDNA-Fl/EB in the PC and solution respectively. [152] Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

4.2. PCs Used for Solar Cell PCs can selectively modify propagation of the light with specific wavelength based on its intrinsic periodic structure. Various photonic devices have been fabricated based on PC structure and light manipulation properties, such as light waveguide that causes light to curve at acute angles,[153-154] dielectric mirrors[155] and so on. In rescent years, scientists have coupled PC structure into dye-sensitive solar cell (DSSC) and investigated its influence on the cell performance.[156-162] Typically, Mallouk et al. pioneered to couple TiO2 PC layer into TiO2 photo-electrode, which brought about an enhancement of light harvesting efficiency due to light localization properties of PCs.[156-157] Miguez’s group carried out series of theorectical and experimental researches to illustrate the light-harvesting enhancement of PCs on the DSSC.[158-160] Huisman et al.[161] replaced the conventional TiO2 photo-anode by

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its inverse opal. However, the procedure gave rise to an obvious drop of the cell efficiency due to difficult fabrication of PC anode with large area. Song et al.[70] developed a high efficient output of DSSC with a PC concentrator (Figure 14). The PC concentrator was developed by combining the advantages of light-focusing of concave-mirror and wavelengthselective filtration of PCs. The as-prepared PC concentrators could effectively improve the output power of the DSSC by more than 5 times by converging the desired light (i.e., the absorbance of the dye) and filtering out harmful IR and UV light, which demonstrated a promising strategy to promote the DSSCs’ practical application.

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Figure 14. The scheme of the photovoltaic system with a PC concentrator. The insert is a typical SEM image of the PC concentrator; (b) The typical photographs of the PC concentrators with different colours, the stopbands of the PC concentrators from left to right are 448, 475, 530 and 647 nm, respectively. The scale bar is 2 cm. (c) The incident photon-to-current conversion efficiency curve of the DSSC, insert numbers are the Pmax of DSSC when using PC concentrators with stopband of 530 nm and control sample. [70] Reproduced by permission of The Royal Society of Chemistry.

4.3. PCs Used for Enhanced Fluorescence PCs have been applied for improving the fluorescence efficiency in many optic devices. The photoluminescence enhancement by PCs has been reported by photonic defects[163] and minimizing surface recombination losses of quantum dots at low temperature.[164] Recently, Klimov et al.[165] have reported amplified spontaneous emission in semiconductor nanocrystals uniformly coated on opal PS surfaces by enhancing the optical gain which was achieved by reducing the group velocity at the edge of the photonic stopband. Vos[166] and Cunningham et al. [167] have successfully realized a five-fold enhancement of the signal-tonoise ratio by utilizing resonant PC-enhanced fluorescence in a cytokine immunoassay. Song et al.[168-169] propose a simple, effective and practicable strategy to enhance fluorescence emission of molecular materials in the solid state by the fabrication of a Bragg mirror of PCs, where, the stopband can inhibit light propagation in a certain direction for a given frequency, therefore, PCs can modify significantly the emission characteristics of embedded opticallyactive materials (dyes, polymers semiconductors, etc.) as the emission wavelengths of the active materials overlap the stopband. An about 20 times enhancement was obtained when using the PC surface (Figure 15a). They recently amplified the fluorescent contrast by PCs in optical storage by applying PCs to the optical storage system.[170] A 40-fold enhancement of the fluorescence signal and a sevenfold ON/OFF ratio amplification are achieved.

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Furthermore, a fluorescence image on the matched PC surface displays both higher brightness and contrast compared with that on glass(Figure 15b). Thus, amplification of fluorescent contrast on the PC surface can provide much better sensitivity and resolution in the bistable photoswitching, which will open a simple and viable way for design and development of high-performance optical memory devices. Song used PCs to enhance the emission density of organic dyes[171] and to achieve nonlinear emission.[172] An high-performance light emitted device (LED) system was obtained when coating PCs upon the surface of LED, which greatly amplify the take-out efficiency of light.[173] Recently, Song [174] synthesized high- photocatalytic performance Ti-Si oxides PCs with hierarchically macro-/mesoporous structure by combination of Polymer PC template and amphiphilic triblock copolymer. It was found that the photodegradation efficiency of i-Ti-Si PCs was 2.1 times higher than that of TiO2 PCs, and a maximum enhanced factor of 15.6 was achieved in comparison to nanocrystalline TiO2 films when the energy of slow photon [175-176] was optimized to the abosorption region of TiO2, which originated from the synergetic effect of slow photon enhancement and high surface area. The slow photon effect of PCs has been applied to improve photo-conduct properties of semiconductor [177] by ozin et al.

Figure 15. (a) The fluorescence spectra of 20 nm thick RB deposited on yellow PCs, aluminium film and glass substrate (λex = 550 nm). Here the fluorescence spectra were collected at the stopband direction of PCs. The inset shows the fluorescence spectra of RB (20 nm) on aluminium film and glass. [168] Reproduced by permission of The Royal Society of Chemistry. (b) Fluorescence intensity associated line profiles from the glass and PC surfaces, respectively. [170] Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

5. Future Outlook PCs have attracted increasing attention due to their special light manipulation properties. They show promising applications in ultrasensitive detector, fluorescence amplification, optic waveguide in various optic device, photo-catalytic, etc. However, there is still a long way for the practical application of PCs due to the difficult fabrication of PCs with large-scale and high quality. Moreover, it is difficult for the fabrication of full-stopband polymer PCs due to the low refractive index of polymer, which greatly restricts wide application of PCs.

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Otherwise, further developing light-manipulation application of PCs in new system, such as detecting minute amount of harmful gas or solid, will be an important research trend for PCs.

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[174] Liu, J., Li, M. Z., Wang, J. X., Song, Y. L., Jiang, L., Murakami, T. & Fujishima, A. (2009). Hierarchically Macro-/Mesoporous Ti-Si Oxides Photonic Crystal with Highly Efficient Photocatalytic Capability. Environmental Science & Technology, 43(24), 9425-9431. [175] Chen, J. I. L., Loso, E., Ebrahim, N. & Ozin, G. A. (2008). Synergy of slow photon and chemically amplified photochemistry in platinum nanocluster-loaded inverse titania opals. J. Am. Chem. Soc., 130(16), 5420-5421. [176] Chen, J. I. L. & Ozin, G. A. (2008). Tracing the Effect of Slow Photons in Photoisomerization of Azobenzene. Adv. Mater, 20(24), 4784-+. [177] O'Brien, P. G., Kherani, N. P., Zukotynski, S., Ozin, G. A., Vekris, E., Tetreault, N., Chutinan, A., John, S., Mihi, A. & Miguez, H. (2007). Enhanced photoconductivity in thin-film semiconductors optically coupled to photonic crystals (vol 19, pg 4117, 2007). Adv. Mater, 19(24), 4326-4326.

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In: Photonic Crystals Editor: Venla E. Laine, pp. 31-66

ISBN: 978-1-61668-953-7 © 2010 Nova Science Publishers, Inc.

Chapter 2

ACHIEVING COMPLETE BAND GAPS USING LOW REFRACTIVE INDEX MATERIAL Dahe Liu1, 2,*, Tianrui Zhai2 and Zhaona Wang2 1

Key Laboratory of Nondestructive Test (Ministry of Education), Nanchang Hangkong University, Nachang 330063, China. 2 Applied Optics Beijing Area Major Laboratory, Department of Physics, Beijing Normal University, Beijing 100875, China

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1. Introduction Photonic crystals (PCs) were introduced by E. Yablonovitch [1] and S. John [2] in 1987. In 1990, Ho et al. demonstrated theoretically that a diamond structure possesses complete band gaps (CBGs) [3]. Since then, great interest was focused on fabricating threedimensional (3D) photonic crystals (PCs) in order to obtain CBGs. Several methods were reported, and CBGs were achieved in the range of microwave or submicrowave. Theoretical analysis showed that although CBGs can be obtained by diamond structure, a strict condition should be satisfied, i.e., the modulation of the refractive index of the material used should be larger than 2.0 [4,5]. So, attention was paid to finding the materials with high refractive index. Some scientists tried to fill the templates with high refractive index materials to increase the modulation of the refractive index [6-8], and CBGs were obtained. However, the CBGs achieved in 3D PCs were mostly in microwave or infrared regions [7,9-11]. Holography is a cheap, rapid, convenient, and effective technique for fabricating 3D structures. In 1997, holographic technique was introduced for fabricating the face centered cubic (fcc) structure [12]. Campbell et al. actually fabricated the fcc structure with holographic lithography [13]. Several authors reported their works on this topic [4,5,9,14]. Toader et al. also showed theoretically a five-beam “umbrella” configuration in the synthesis of a diamond photonic crystal [15]. Because both the value and modulation of the refractive index of the holographic recording materials are commonly low, there would be no CBGs in PCs made directly by holography. For example, the epoxy photoresist generally used for holographic lithography *

E-mail address: [email protected]. (Corresponding author: Dahe Liu)

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Dahe Liu, Tianrui Zhai and Zhaona Wang

[13,16-18] has n=1.6, which is a little bit too low. They may, however, be used as templates for the production of inverse replica structures by, for example, filling the void with high refractive index and burning out or dissolving the photoresist [13], and a good work was done by Meisel et al [19,20]. However, to find materials with large refractive index is not easy, and the special techniques needed are very complicated and expensive. It limits the applications of PCs, especially for industrial productions. Although some efforts had been made [20], CBGs in the visible range had not yet been achieved by using the materials with low refractive index. Therefore, it is a big challenge to fabricate 3D PCs possessing CBGs in the visible range by using materials with low refractive index, though it is greatly beneficial for the future PC industry. As a first step for achieving this, it is important to obtain very wide band gaps. In the previous investigations [21,22], it was evidently shown that the anisotropy of a photonic band gap in a two-dimensional photonic crystal is dependent on the symmetry of the structure, and as the order of the symmetry increases, it becomes easier to obtain a complete band gap. One would naturally ask whether or not such a method is applicable to 3D PCs. In view of this question, we proposed a series holographic method for fabricating some special structures by using materials with low refractive index, and the features of the band gap in such structures were then studied experimentally and theoretically.

2. Complex Diamond Structure [23]

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1) Experimental Method and the Samples It is known that a cell of diamond structure consists of two cells of fcc structures, and the two cells have a distance of one-quarter of the diagonal length of the cell along the diagonal line. According to this, the diamond structure here was implemented by holography through two exposures: an fcc structure was recorded in the first exposure, then, after the recording material was translated one-quarter of the diagonal length along the diagonal line of the fcc structure, a second exposure was made to record another fcc structure. In this way, a PC with diamond structure was obtained. Figure 1 shows schematically the optical layout in our experiments. Four beams split from a laser beam were converged to a small area. The central beam was set along the normal direction of the surface of the plate, while the other three outer beams were set around the central beam symmetrically with an angle of 38.9° with respect to the central one. The laser used was a diode pumped laser working at 457.9 nm with linewidth of 200 kHz (Melles Griot model 85-BLT-605). The polarization state of each beam was controlled to achieve the best interference result [24,25]. The recording material was mounted on a one-dimensional translation stage driven by a stepping motor with a precision of 0.05 μm/step. The translation stage was mounted on a rotary stage. The holographic recording material used was dichromated gelatin (DCG) with refractive index n=1.52. The maximum value of its refractive index modulation Δn can reach around 0.1, which is a very small value for obtaining wide band gaps. The thickness T of the material was 36 μm. The DCG emulsion was coated on an optical glass plate with flatness of λ/10 and without any doping.

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33

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It is known that there are several directions with high symmetry in the first Brillouin zone of an fcc structure (see Figure 2). In our experiments, two exposures as mentioned above were made firstly in Γ-L ([111]) direction to get a diamond structure. Then, a second, even a third, diamond structure was implemented by changing the orientation of the recording material by rotating the rotary stage to other symmetric directions, i.e., the direction of Γ-X ([100]) and/or the direction of Γ-K ([110]) in the first Brillouin zone of the first fcc structure. In this way, photonic crystals with one, two, and even three diamond structures were fabricated. It should be pointed out that when the second or the third diamond structure was recorded, the angle between any two beams should be changed to guarantee that all the beams inside the medium satisfy the relation shown in Figure 1 so that the standard diamond structures can be obtained. Since the hologram made with DCG is a phase hologram, there is only the distribution of the refractive index but no plastic effect inside the hologram, so a scanning electron microscope (SEM) image cannot be obtained. However, microscopic image can be obtained. To verify the structure in the hologram, the same fcc structure was implemented using the same optical layout but with the photoresist of 2 μm thickness. Figure 3 shows the structure made with the photoresist at [111] plane: Figure 3 (a) shows the SEM image and Figure 3(b) shows the optical microscopic image taken with a charge coupled device (CCD) camera mounted on a 1280× microscope. The pixel size of the CCD is 10 μm. Figure 3(c) shows the optical microscopic image of a diamond structure made with DCG of 36 μm at [111] plane, which was also recorded by the CCD camera mounted on a 1280× microscope as mentioned above.

38.9°

K2

K3

K4 K1 G2 G3 G1 Figure 1. Schematic optical layout for recording an fcc structure.

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Dahe Liu, Tianrui Zhai and Zhaona Wang

Figure 2. First Brillouin zone of an fcc structure.

(b)

(a)

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(c) Figure 3. Structure of PCs made in our laboratory. The images (b) and (c) were recorded by a CCD camera mounted on a 1280× microscope. (a) SEM image of an fcc structure made with photoresist of 1 μm thickness. (b) Microscopic image of an fcc structure made with photoresist of 1 μm thickness. (c) Microscopic image of a diamond structure made with DCG of 36 μm.

The theoretical calculated diamond lattice formed by two exposures as mentioned above is shown in Figure 4. It should be pointed out that the shape of the interference results appears like an American football. In this figure, the vertical bar shows a bottom to top gradient corresponding to the outer to inner region of a cross-sectional cut of a football. The gray gradient along the outer surface is related to the value of the vertical bar. Whether a diamond lattice can be obtained depends on how far the two footballs stand apart. If the two footballs are large enough that they overlap, the structure cannot be considered as a diamond lattice. If the two footballs can still be recognized as two, it means that a diamond lattice is formed. In our experiments, the absorption coefficient α of the material was chosen as α = 1/ T (Ref. 26) to get optimal interference result. According to previous works, it is well known that for holographic recording materials the relation between the optical density and the exposure is nonlinear. The holographic exposure can be controlled in the region of stronger nonlinear dependence between optical density and exposure as discussed previously [27]. In this way,

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two fcc lattices can be recognized, so that the structure can be considered as a diamond lattice.

Figure 4. Diamond lattice formed by two fcc lattices in nonlinear exposures. The vertical bar shows a gradient of bottom to top corresponding to the outer to inner region of a cross-sectional cut of a football. The gradient of the outer surface is related to the value of the vertical bar.

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Collimated white light beam

Right triangular Prism n=1.5 Glass substrate n=1.5 DCG material n=1.52 (average) To spectrometer

Figure 5. Setup geometry for measuring transmission spectra of photonic crystals.

The transmission spectra of PCs with diamond structure made with DCG were measured. In the measurements, a J-Y 1500 monochromator was employed. To minimize the energy loss from reflection at large incident angle, several right triangular prisms were used. The measuring setup geometry is shown in Figure 5. The measured range of the wavelength was 390–800 nm. The [111] plane of the PC measured was set firstly at an arbitrary orientation. The incident angle of the collimated white light beam was changed from 0° to ±90° (the incident angle is in the plane parallel to the surface of the paper). Then, the PC was rotated to other orientations (the orientation angle is in the plane perpendicular to the surface of the paper), and the incident angle of the collimated white light beam was also changed from 0° to ±90°. The orientation angle and the incident angle are the angles ϕ and θ in a spherical system indeed. In this way, the measurements give actually 3D results.

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Dahe Liu, Tianrui Zhai and Zhaona Wang

2) Measured Spectra and Discussions

0.4

Transmission

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The measured transmission spectra of a fabricated triple diamond PC are shown in Figs. 6(a)–6(c), denoting measured results at different orientations of the sample. Each curve in the figures gives the spectrum measured at a certain incident angle and a certain orientation. Different curves give the results at different incident angles and orientations. Thus, whether or not the common gap exists can be determined by the intercept of all the curves. It can be seen that, for the PC with three diamond structures (six fcc structures), the width of the band gaps reached 260 nm, the ratio between the width and the central wavelength of the gaps reached 50 %, and there is a common band gap with a width of about 20 nm at 450–470 nm in the range of 150° of the incident angle. The common gap obtained using DCG with very low refractive index (n=1.52) existed in a wide range, which reached 83 % of the 4π solid angle. Although a complete band gap for all directions was not obtained in our experiments, it is significant to achieve such a wide angle band gap by using a material (DCG) with very low refractive index, because this angular tuning range satisfies most applications in practice, for example, restraining spontaneous radiation with low energy loss in a wide range, wide angle range filter, or reflector with low energy loss. This interesting result comes from a complex diamond structure. The photonic crystal with triple-diamond structures is an actual multi structure, but not a stack of several same structures. The [111] direction of the first diamond structure is actually the Γ − X direction of the second diamond structure and the Γ − K direction of the third diamond structure, respectively. When a beam is incident normally on the PC, the beam is in the [111] direction of the first diamond structure. When the incident angle changes, the beam deviates from the [111] direction of the first diamond structure and tends to approach the [111] direction of the second or third diamond structure. Therefore, though the incident angle changes, the light beam remains always around the [111] direction of the other diamond structures. The narrow region in the K − ω dispersion relation of a diamond lattice may be expanded by other diamond lattices.

0.3

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(a) Figure 6. Continued on next page.

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0.25

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0.05 0.00 400

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(c) Figure 6. Measured transmission spectra of PC with three diamond lattices recorded by holography in Γ-L, Γ-X, and Γ-K directions of an fcc lattice respectively. (a), (b), and (c) correspond to the measured results. The angles appearing in (a), (b), and (c) are the incident angles, which are in a plane parallel to the surface of the paper. (a) Orientation angle is 0°. (b) Orientation angle is 30°. (c) Orientation angle is 90°.

The advantage of a multi diamond structure can be seen more clearly from a comparison with those having one or two diamond structures. Figures 7 and 8 give the measured transmission spectra of PCs with one and two diamond lattices, respectively, at all orientations. The structure with single diamond lattice was fabricated in Γ − L [111] direction, while the structure with two diamond lattices was fabricated in Γ − L [111] and Γ − X [100] directions. Besides, the method used for measuring the two structures with single and double diamond lattices, respectively, was similar to that mentioned above. It can be found that, for the single diamond structure, the space angle range of the common gap is 40°; for the double diamond structures, it becomes 80°. Comparing Figs. 6–8, it is obvious

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Dahe Liu, Tianrui Zhai and Zhaona Wang

that the width of the band gaps of a PC can be broadened effectively by increasing the number of diamond lattices. This means that the common gap can be enlarged by means of multi structures. The physical origin of such a phenomenon can be understood from the change of structure symmetry. With the increase of the number of diamond lattices, the symmetries around the center of the structure become higher. This made it easier to obtain a broader response in many directions (common band gap in a wide range of angles), which is similar to the case of two dimensional PCs (see Refs. 19 and 20).

 

0.7

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(a)

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0.7

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(b) Figure 7. Continued on next page.

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0.7

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(c) Figure 7. Measured transmission spectra of PC with single diamond lattice recorded by holography in Γ − L direction of an fcc lattice. (a), (b) and (c) correspond to the measured results. The angles appeared in (a), (b) and (c) are the incident angle which is in a plane parallel to the surface of the paper. (a) Orientation angle is 0° . (b) Orientation angle is 30° . (c) Orientation angle is 90° .

0.5

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0.6

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(a) Figure 8. Continued on next page.

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Dahe Liu, Tianrui Zhai and Zhaona Wang

0.5

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0.0 400

500

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(c) Figure 8. Measured transmission spectra of PC with 2 diamond lattices recorded by holography in Γ − L and Γ − X directions of an fcc lattice respectively. (a), (b) and (c) correspond to the measured results. The angles appeared in (a), (b) and (c) are the incident angle which is in a plane parallel to the surface of the paper. (a) Orientation angle is 0°. (b) Orientation angle is 30°. (c) Orientation angle is 90°.

3) Conclusion The width of the band gaps in a PC made by a material with low refractive index can be broadened greatly by multi diamond structures, and a common band gap in the range of 150° of the incident angle can be obtained. This technique will be greatly beneficial in achieving complete band gaps by using materials with low refractive index.

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41

3. Self-simulating Structure [28,29]

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As mentioned above, a complex diamond lattice was fabricated using materials with low refractive indices. Although a common gap in the visible range was obtained in a wide range that reached 83 % of the 4π solid angle [23], the CBGs were not yet obtained. However, regardless of whether or not they succeed, the above-mentioned techniques are complicated and difficult to carry out. Now, a novel technique has been developed by which a wide CBG can be achieved easily in the visible range using materials with a low refractive index. It is obvious that, for a self-simulating spherical system shown in Figure 9, when a light beam passes through the center of the sphere, it will be symmetric in all directions with respect to the center. Therefore, if a PC has this self-simulating spherical structure, it will possess complete band gaps. It should be emphasized that achieving CBGs using this structure does not pose any special requirement on the refractive index of the material used. This type of self-simulating spherical structure can be implemented easily via holography. This includes three steps: 1) preparation of the recording material by coating the emulsion on a soft substrate; 2) fabrication of a 1D PC in rectangular coordinates by using this recording material; 3) to make this PC into a sphere. In this way, a 1D PC in rectangular coordinates was transformed to a spherical 1D PC system.

Figure 9. Self-simulating spherical structure.

Figure 10 shows the experimental set-up geometry. Dichromated gelatin (DCG) was still used as recording material. The recording material was prepared by coating a thick layer of 36 mm dichromated gelatin (DCG) onto polyethylene terephthalate (PET). The refractive index of the DCG is 1.52, and the refractive index of the PET is 1.64. Figure 6 shows the optical layout. The laser used was a diode-pumped laser working at 457.9 nm with a line width of 200 KHz (Melles Griot model 85-BLT 605). The monochrometer used was a J–Y 1500. The light beam for measuring was a narrow beam with a diameter of 3 mm. By controlling the wavelength of the laser and the angle between the two beams incident on both sides of the recording material, the period of the PC could be changed for working in the visible, IR, or UV range. Because of the influences of the absorption by the recording material [30], and some factors in processing after exposure of the recording material [31], the structure of the PC in rectangular coordinates is not strictly periodical. Figure 11 shows the structure and distribution of the modulation Δn of the refractive index inside the PC.

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Dahe Liu, Tianrui Zhai and Zhaona Wang

When the PC in rectangular coordinates is transformed to a sphere, its structure is just the self-simulating spherical system shown in Figure 9. Several self-simulating spherical structures with different diameters were fabricated. All of them showed identical characteristics under the same experimental conditions. The measured transmission and reflection spectra of a self-simulating spherical structure with the radius of 20 mm were demonstrated as follows. During the measurements, the angles θ and ϕ of the spherical system were changed from 0° through 180° and from 0° through 360° , respectively. Figs. 12 and 13 give only the measured results for ϕ changes from 0° through 360° while θ was set at a fixed value of 90° . When the value of θ is changed in the range of 0 − 180° , the measured transmission and reflection spectra are quite consistent. For comparison, the theoretical results calculated by transfer matrix method are also shown. Laser 

P

M

L 

Recorder 

 P       computer 

L  monochrometer

BS 

M

L  M 

M

S L

M  L

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E           (a) Fabricating set‐up geometry                                  (b) Measuring set‐up geometry 

PC

Figure 10. Optical layout for fabricating and measuring PC. L is lens, M is mirror, BS is beam splitter, E is recording material, S is white light source, PC is photonic crystal, P is photomultiplier tube.

Figure 11. Structure of the fabricated PC in rectangular coordinates.

From Figure 12, it can be seen clearly that, for different values of w in the range of 0– 360°, the measured transmission spectra almost completely overlapped with little frequency shift. The width of the CBG reaches 120 nm although the refractive index of DCG is pretty low. The reason is that a structure with non-strict period can broaden the band gaps

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effectively [32]. Figure 13 shows that the reflection spectra are just complementary to the transmission spectra. In practical experiments and applications, frequently, the incident beam may not pass through the center of the self-simulating spherical structure strictly. It may affect the characteristics of the band gaps. Figure 13 show the measured transmission spectra when the incident beam deviates from the center of the self-simulating spherical structure. It can be seen that, even though the deviation is as large as 15 mm from the center of the sphere with a radius of 20 mm, there is still a complete band gap with a width of 30 nm. This deviation corresponds to the condition that a light beam incidents on the structure with an incident angle of 49°.

(a) Measured spectra.                                  (b) Calculated spectra.

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Figure 12. Measured and calculated transmission spectra of a self-simulating spherical structure for different value of ϕ at θ = 90° .

(a) Measured spectra

(b) Calculated spectra.

Figure 13. Measured and calculated reflection spectra of a self-simulating spherical structure for different value of ϕ at θ = 90°

Self-simulating spherical structures with radii of 5 mm, 7.5 mm, 10 mm, 20 mm, and 25 mm were fabricated and measured in our laboratory, and similar results to those shown in Figs. 12 and 13 were obtained. The measured transmission and reflection spectra revealed that a true complete band gap actually exists for this kind of structure. The fact that the light

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Dahe Liu, Tianrui Zhai and Zhaona Wang

was forbidden from passing through the structure is really caused by the band gaps, rather than induced by other physical mechanisms, for example, total inner reflection and absorption. An actual fabricated Self-simulating spherical structure is shown in Figure 14. In summary, a self-simulating spherical structure possesses complete band gaps. This kind of structure can be implemented easily in terms of holography. Achieving complete band gaps with this structure can use common holographic recording materials with low refractive indices rather than high refractive index materials. Even though the working condition has a relative large deviation from ideal conditions, there is still a complete band gap.

4. A Dnv Point Group Structure Based on Heterostructure [33]

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Although the self-simulating spherical structures can obtain CBG, the incident beam must pass through the center of the sphere. however, when light incidents on the surface of the sphere with a incident angle larger than 49°, the CBG will disappear. This disadvantage limits the application of the structure, such as the use of wide incident beam in practical applications. So, improvement of the self-simulating spherical structures is needed. It was demonstrated that PC heterostructure can improve effectively the characteristics of PCs, and can provide a necessary variation in material properties to turn a photonic crystal raw material into a functional device [34] by means of self-organizing [35-37], lithography with electron beam [38,39] or autocloning [40]. PC heterostructures, like their semiconductor counterparts which are made by combining at least two materials that have distinct band structures, can be fabricated by changing the lattice constant, hole size, or even lattice geometry in the crystal. This can be done either abruptly or gradually [34].

Figure 14. Photographs of cut piece of 1D PC in rectangular coordinates and PC in self-simulating spherical structure

Now, a new technique is proposed: we first developed a technique to fabricate a heterostructure with gradual refractive index distribution and gradual period, and obtained 2D ODBG. Further, a self-simulating sphere structure was implemented based on this kind of heterostructure, and a genuine CBG in discretionary condition can be achieved. Figure 15 shows the set-up geometry for fabricating our heterostructure. The laser used was a DPSS laser working at 457.9 nm with line width of 200 kHz (Melles Griot model 85BLT-605). The holographic recording material was dichromated gelatin (DCG). The DCG

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emulsion of 36 μm thickness was coated on an optical glass plate with flatness of λ/10 and without any doping. The plate was mounted on a 1D translation stage driven by a stepping motor with a precision of 0.05 μm/step. For an elementary reflection hologram made by two beams with 180°angle, because of the absorption by the recording material the distribution of refractive index inside the medium is a hyperbolic cosine function [30] (see Figure 16), but neither strictly sinusoidal nor strictly period. Besides, swelling of the DCG emulsion exists during the processing after exposure, and the swelling is not uniform [31]. So, we introduced a linear expansion parameter Mz, and the thickness swelling of jth layer at position z can be expressed as [31] M z , j = M 1 − ( M 1 − M N ) j / N + rj / a

(1)

where Mz, j is the expansion parameter for jth period, rj is a random number in the range

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[0,1], N is the number of layers, parameter a controls the amplitude of the random fluctuation. The linear random expansion is shown in Figure 17 [31]. So, the final result actually likes a chirped grating with a variable period.

Figure 15. Set-up geometry of triangular lattice. The angles between E1 and E2, E2 and E3, E1 and E3 are all 120°.

Figure 16. Schematic diagram of distribution of refractive index inside an element reflection hologram. T is the thickness of the material.

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Dahe Liu, Tianrui Zhai and Zhaona Wang

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Figure 17. Linear random expansion series (Ordinate) of holographic recording material during the processing after exposure. Abscissa represents period numbers.

The results in Figures 16 and 17 show that what we obtained for an elementary reflection hologram is actually a heterostructure, and the interfaces of the heterostructure are gradual surfaces rather than step functions. In Figure 16, T is the thickness of the material. Also, the period is gradually changed rather than a constant. Therefore, the PC made by the method shown in Figure 15 is a PC heterostructure. In our experiments, two exposures were taken for fabricating triangular lattices using the set-up geometry shown in Figure 15. After the first exposure, the recording material was translated a distance of d/3 along the Γ − K direction of the first Brillouin zone, and then the second exposure was made. In this way, the lattice-point shape is not circular, but elongated and became elliptical. Figure 18 shows the transmission spectra of the triangular lattice heterostructure at different incident angles. It can be seen clearly that there is an absolute band gap in the wavelength range of 430 nm-500 nm at different incident angles, the width (FWHM) of the gap is about 50 nm (445-495 nm). It is an ODBG in deed. The oscillation at the bottom of the band gap is induced by the nonconforming periodicity. The above experimental results could be explained theoretically as follows.

Figure 18. Transmission spectra of triangular lattice heterostructure made by two exposures with a translation. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

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For the dielectric columniation triangular lattice, the first band and the second band of TM wave at K of its first Brillouin zone is degenerated because of high symmetry. It results in that there is no common gap for TE and TM waves in Γ − K direction. In our experiments, the method of two exposures with a translation was used, it made the lattice point shape to be non-circular (see the left iconograph at the bottom in Figure 19). So, the symmetry point group of this kind of triangular lattice is debased to C2v from C6v. It releases the band degeneration of TM wave at K of the first Brillouin zone, and this property is shown clearly in Figure 19 (pointed by the black arrows 1-4). The non-circular lattice-point shape of the triangular lattice induces two extra symmetric points K1 and M1 (see the right iconograph at the bottom in Figure 19) The gradual period of the heterostructure is an important factor for obtaining ODBG. In this kind of structure the lattice constant is changed gradually and linearly. According to our experimental conditions, in our fabricated structure, there are 150 layers, they can be treated as 150 sub-structures, the minimum lattice constant is d=234.8 nm. During the processing of DCG after exposure a few air holes may be formed by dehydration, considering the refractive index of DCG be 1.52 before exposure, we choose the minimum refractive index n0 = 1 . The distribution of refractive index is proportional to light intensity and can be expressed by

n = n0 +

I=

∑E

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s =0 , a / 3

2 0

I max

I Δn − I min

,

(2)

⎡ ⎛ 3 ⎞ ⎛3 ⎛ 3 ⎞⎤ ⎞ ky ⎟ cos ⎜ k ( x + s ) ⎟ + 4 cos 2 ⎜ ky ⎟⎥ ⎢1 + 4 cos ⎜⎜ ⎟ ⎝2 ⎜ 2 ⎟⎥ 2 ⎠ ⎢⎣ ⎝ ⎠ ⎝ ⎠⎦

(3)

where, s is displacement quantity along Γ − K direction of the first Brillouin zone. The permittivity is

ε = n + 2n0 Δn ⋅ 2 0

I max

⎞ ⎛ I I Δn ⎟⎟ + ⎜⎜ − I min ⎝ I max − I min ⎠

2

(4)

where, Δn = 0.5 , I max − I min = 9 . Because the heterostructure is a gradual period structure consist of 150 sub-structures, the upper band edge of the first sub-structure and the bottom band edge of the last (N=150) substructure determine the width of the total band gap of the heterostructure. In our calculations, the structure with the minimum period d (the first sub-structure) and that with the maximum period 1.15 d (the Nth sub-structure) were taken for calculations so that the upper and the bottom band edges can be obtained. Figure 19 gives the results calculated by plane wave expansion method. The dark solid line and the dark dots represent the bands of the first substructure for TM and TE waves respectively. The upper edges of the two bands form the high frequency edges of the heterostructure for TM and TE waves. The light dashed line and the light dots represent the bands of the Nth sub-structure for TM and TE waves respectively, The bottom edges of the two bands form the low frequency edges of the heterostructure for TM

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Dahe Liu, Tianrui Zhai and Zhaona Wang

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and TE waves. The total band gap of the heterostructure is the overlap of the band gaps of the 150 sub-structures. In any high symmetric direction of the first Brillouin zone, the directional band gap of the two sub-structures form the upper and the bottom band edges of the total band gap of the heterostructure in this direction respectively (as shown in light grey zone). The dark grey zone is the common gaps of all directional gaps so that an ODBG of the heterostructure is formed. The calculated position of the band gap locates at 3.8×1015～4.1×1015 s-1 (460nm～496nm), they (and the width) are well consistent with the experimental results.

Figure 19. Calculated dispersion relation of gradual triangular lattice heterostructure. The left iconograph at the bottom shows the lattice geometry. The right iconograph at the bottom shows the first Brillouin zone of the triangular lattice.

One of the characters of this kind of triangular lattice is that the bands of TE and TM waves are almost superimposed, and along every direction band gaps are close to each other. This character is very helpful for achieving ODBG through gradual period heterostructures, when the layers of the heterostructure is more enough, and the expansion of the recording material reaches a certain range, a wider ODBG will be obtained by use of narrow directional band gaps. Obviously, it will be helpful for further works to achieve 3-D complete band gaps. Based on the principle in Ref. [28] this kind of heterostructure can be transformed into a self-simulating spherical structure in the following steps: Firstly, preparing holographic recording material by coating 36 μm thick DCG emulsion on soft substrate polyethylene terephthalate (PET); Then, fabricating gradual index distribution and period heterostructure using the recording material; Finally, transforming the heterostructure into self-simulating sphere. This self-simulating sphere structure can be expressed by Dnv point group. Figure 20 shows a Dnv self-simulating sphere structure based on the gradual index distribution and gradual period heterostructure.

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36μm

(a) Schematic diagram of DnV self-simulating sphere structure.

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(b) The schematic diagram of the cut slice of the sphere shown in (a) Figure 20. Structure of DnV point group.

Figure 21. Measured transmission spectra of a D12V self-simulating sphere structure for different value of ϕ at θ = 90° . Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

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The transmission spectra of a D12v self-simulating sphere structure was measured and shown in Figure 21. In the measurements, a parallel beam with a diameter of 1 mm was used as the incident light, and was incident on the surface of the sphere passing through the center of the sphere. It is formed by a collimated beam passing through a 1 mm diaphragm. During the measurements, the angles θ and ϕ of the spherical system were changed from 0° through 180° and from 0° through 360° respectively. In our experiments, several D12v selfsimulating sphere structures with different diameters were fabricated. All of them showed identical characteristics under the same experimental conditions. It can be seen obviously that, for incident beams from different ϕ at θ = 90° , the position of the band gap kept almost unchanged, and its width reaches about 70 nm.

y

α R

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x

Figure 22. Schematic diagram when the incident light deviates from the center of the sphere. The corresponding incident angle is α = arcsin (l R ) .

Figure 23. Measured transmission spectra of a D12v self-simulating sphere structure for different values of ϕ at θ = 90° when the incident beam shifts different distances from the center. The radius of the sphere is 12.5 mm. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

Achieving Complete Band Gaps Using Low Refractive Index Material

51

Besides, when the incident light is translated from the center of the sphere for different shifts l (see Figure 22). the transmission spectra are measured (see Figure 23). In the measurements, the radius of the sphere was 12.5 mm. It can be found that, even the shift is as large as 12.3 mm from the center of the sphere, there still is a CBG with a width of 40 nm. This shift corresponds to the condition that a light beam incidents on the surface of the sphere with an incident angle larger than 80° . In our experiments, two D12v self-simulating sphere structures were superimposed orthogonally to improve further the property of the band gap. In summary, a Dnv point group structure based on 2-D gradual index distribution and gradual period heterostructure and self-simulating sphere possesses genuine complete band gap in a discretionary condition. The Dnv self-simulating sphere structure can be implemented easily by holography using low refractive index materials.

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5. Theoretical Investigation [41] The band gaps of PC were described with various methods, such as plane wave expansion method (PWEM) [3,42,43], transfer matrix method (TMM) [44,45], multiscattering theory [46], tight-binding formulation [47] and finite-difference-time-domain (FDTD) method [48,49]. However, only numerical solutions were obtained by these methods, in which their physical images are not intuitionistic or not clear enough. Therefore, a concise analytical solution (AS) will be valuable for analyzing more thoroughly the properties of the band gaps. Besides, these methods were mostly used for calculating the problems of PCs with step distribution of the refractive index. Recently, the present authors obtained very wide 3-D band gaps [23], even complete band gaps [28,33], using holographic technology and a low refractive index material basing on the principle of multi-beam interference [12,13], and has been used widely, only that the distribution of refractive index inside the recording material (such as photopolymers [50], liquid [51,52], or dichromated gelatin [23,28,33]) is gradual function rather than step function. Liu et al. [53] derived an analytical solution for 1-D PC with a sinusoidal distribution of refractive index. Samokhvalove et al. [54] made an approximation analysis of 2-D PC with rectangular dielectric rods. Nusinsky et al. [55] also made analytical calculation of rectangular PC. Analytical solutions presented by [54] and [55] are valid only for rectangular, step index photonic crystals. While the solution presented in this paper is valid for different structure having gradual distribution of refractive index. Now, the theoretical analysis is investigated for the band gap characteristics of the PC made by multi beam interference. An analytical solution can be obtained

(1) Basic Considerations For multi-beam interference, the intensity distribution is a gradual function. When the distribution of the refractive index inside the recording material is proportional to the interference intensity, it should keep in consistency with the gradual distribution according to the property of interference. Therefore, only a few non-zero low order terms exist in the Fourier expansion of the dielectric constant. Since the wave function is generally a coupled multivariate linear equation group, a clear and simple AS can be obtained by approximation according to the spectral property.

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Dahe Liu, Tianrui Zhai and Zhaona Wang

For simplicity, an example of 2-D triangular lattice formed by interference of three coherent beams is shown in Figure 15. In Figure 15, the angle between and is , and points just to the bisector of this angle. Since polarization and phase have no effect on the final result, they were neglected in the following analysis. The interference intensity in the x-y plane is I=

3

∑ Ei

2

i =1

= E02 ⎣⎡1 + 4cos ( ky sin α ) cos ⎣⎡ kx (1 + cos α )⎦⎤ + 4cos 2 ( ky sin α )⎤⎦

(5)

Assuming that the permittivity of the material is proportional to the interference intensity, I 2 after exposure, here, n0 is the minimum refractive index it will be ε = n0 + 2n0 Δn I − I max min of the material, and Δn is the modulation of the refractive index. Its Fourier transform will be

F (ε ) = ( n02 + 3C ) δm,n + C (δm−1,n+1 + δm−1,n +δm,n+1 + δm,n−1 + δm+1,n + δm+1,n−1 )

(6)

2n Δn

0 where, C = I − I min max

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. It can be seen from Eq. (6) that there are only direct current (DC) term and first order 2 term . Obviously, the value of DC term n0 + 3C is much larger (more than 20 times) than that of the first order term. For example, in dichromated gelatin (DCG) [23,28,33], where n0 = 1.52 , Δn = 0.07 , for the triangular lattice formed by the inference of three beams with n02 + 3C ≈ 101 , and all the first order terms are I − I = 9 unit amplitude, we have max min , i.e., C equal to C. So, the coupling among the first order terms could be neglected, and consider only the coupling between the DC term and any one of the first order term. So ε G ′−G can be taken ⎡n02 + 3C C ⎤ ⎢ ⎥ n + 3C >> C could be satisfied when Δn ≤ 0.4 . as ⎢ C n02 + 3C ⎥⎦ . For DCG, the relation 0 ⎣

All the band gaps belong to first Brillouin zone can be obtained by solving a coupled binary linear equation group, and each band gap corresponds to a simple AS.

(2) Analytical Approach (AS) From Maxwell equations and Bloch’law，for TM wave, we have det

{ ( abs (k + G ′) abs (k + G ) )

}

ε G′−G = 0

for TE wave, we have

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(7)

Achieving Complete Band Gaps Using Low Refractive Index Material

{(

}

)

det Re ( k + G ′) Re ( k + G ) + Im ( k + G ′) Im ( k + G ) ε G′−G = 0

53

(8)

Figure 24 shows schematically a reciprocal lattice of a triangular lattice. The solid line represents the first Brillouin zone, and the dashed line represents the cell in the reciprocal lattice. K1 and K2 are two highly symmetric directions of the first Brillouin zone. G1-G6 are the representative reciprocal lattice vectors that collect all values of G and G' in Eqs. (7) and (8) in the cell of the reciprocal lattice. It can be seen that from the geometric relationship in Figure 2, k + G0 = k + G1 , where, k is in K1 and K2 direction respectively.

eˆ y

G4

b2 b1

G2

G6

K2

eˆx K1

G0 G1

G5

G3

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Figure 24. Schematic diagram of the reciprocal lattice of a triangular lattice. The dashed line represents the cell in the reciprocal lattice. The solid line represents the first Brillouin zone

Let a be the lattice constant, so, the vector of its reciprocal lattice are b1 and b2 (see Figure 24). 2π 2π ⎛ ⎛π ⎞ ⎞ 4π eˆ y ⎜⎜ eˆ x + tan⎜ ⎟eˆ y ⎟⎟ , b2 = ˆ For a triangular lattice b1 = a cos(π 6 ) , then we have K1 = 3a ex , a ⎝ ⎝6⎠ ⎠ K2 =

π a

eˆ x +

π 3a

eˆ y

. For TM wave, Eq. (7) can be transformed into

( ) ω (n + 2C)(n + 4C) − c 2

k + G0 n02 + 3C 2 0

2 0

(n + 2C)(n + 4C) k + G (n + 3C) ω (n + 2C)(n + 4C) − c 2 0

2 0

2

− k + G0 k + Gi C

0

(n + 2C)(n + 4C) 2 0

− k + G0 k + Gi C

2

2

2 0

2 0

=0

2 0

2

2 0

2

(9)

The positions of the top and the bottom edges of the first band gap are at

ω± =

(n

(n

2 0

)

+ 3C ± C k + G0 c + 2C n02 + 4C 2 0

)(

)

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(10)

54

Dahe Liu, Tianrui Zhai and Zhaona Wang ⎞

⎛

⎞

⎛

1 2 Δn 1 4 Δn So, ω + ≈ n ⎜⎜1 − 9n ⎟⎟ k c and ω − ≈ n ⎜⎜1 − 9n ⎟⎟ k c can be obtained from Eq.(10) by using the 0

⎝

0

⎠

0

⎝

0

⎠

9 condition n02 + 3C >> C (i,e. Δn ≤ 34 n0 ).

Therefore, the following conclusions can be obtained: 1) The top and the bottom edges of the first band gap of TM wave are proportional to 1 Δn and k ( k ∝ λ , λ is the light speed in the material), and are inversely proportional to n0 . This is more definite than the common knowledge about

ω i + + ωi −

) is negatively related to the average refractive 2 index”, and shows the quantitative dependence of the top and the bottom band edges on material parameters.

“location of the band gap (

2) For a certain material (i.e., n0 and Δn are fixed), the band gaps will be determined only by the value of k . So, the closer the shape of the first Brillouin zone approaches a circle, the smaller the variation of k (i.e.,

k1 ≈ k 2

). In this case,

positions of the directional band gaps in K1 and K2 are almost same, obviously, it is more favorable for the appearance of absolute band gaps. ω i + − ω i − 2 Δn Δω 9 3) The relationship that ω = 2 ω + ω ≈ 9n k c can be achieved (here Δn ≤ n0 ), i+ i− 0 34 i.e. the bandwidth of the first band gap of TM wave is positively related to Δn and

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k , and negatively related to n0 . This is more definite than the common knowledge

“the higher the ratio of refractive indexes, the wider the band gaps”. Figure 25 shows the relationship between the band width Δω ω and n 0 and Δn . In Figure 25, the color bar gives the normalized bandwidth Δω ω , the top line represents the width corresponding to the critical condition Δn = (9 / 34)n0 . It shows significantly that one does not have to seek for high refractive index materials for obtaining wider band gap. Band gap could readily be obtained with lower refractive index materials as same as that obtained with higher refractive index material (under the prerequisite condition of Δn = (9 / 34)n0 ). Large value of the modulation of refractive index needs high refractive index, it is not definitely helpful for obtaining a broad band gap. These features are obviously different from the relationship between the characteristics of band gap and the material parameters of 1-D PC [56]. But, the basic tendencies of variation of the band gap with material parameter obtained from both considerations are similar. Also, for TE wave, Eq. (8) could be transformed into

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Achieving Complete Band Gaps Using Low Refractive Index Material

( ) ω Re[(k + G )(k + G ) ]C (n + 2C)(n + 4C) − c − (n + 2C)(n + 4C) k + G (n + 3C) ω Re[(k + G )(k + G ) ]C (n + 2C)(n + 4C) − c − (n + 2C)(n + 4C) 2

k + G0 n02 + 3C 2 0

∗

2

0

2 0

2

0

i

2 0

2 0

2

∗

2 0

55

0

i

2 0

2 0

2 0

2

2 0

2

=0

(11)

Figure 25. Calculated relationship between band width Δω / ω and n0 and Δn . The color bar gives the band width Δω / ω , the black dash line represents the width corresponding to Δn = (9 / 34)n0 .

The top and the bottom edges of the first band gap are at 2

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ω± = c

(

)

[(

)(

)] ∗

k + G0 n02 + 3C ∓ Re k + G0 k + G1 C

(n

2 0

)(

+ 2C n02 + 4C

)

(12)

Let β be the angle between K1 and K2 in the direction of K1 (see Figure 24), then,

ωi ± =

(n + 3C ) ± C cos 2β (n + 2C )(n + 4C ) 2 0

2 0

2 0

k + G0 c which is consistent with Eq.(10). It shows that the

influencing factors on the width of the first band gap of TE wave, and the rule of their variations are as same as those for TM wave. ∗ ∗ In K2 direction, (k + G1 ) = −(k + G0 ) , then Eq. (8) could be transformed to Eq. (6). This is the reason why same results for TE and TM waves are always obtained in the direction of K2 calculated by PWEM under the condition of small modulation of refractive index.

(3) Comparison with Plane Wave Expansion Method (PWEM) 2 2 It is clear from Eqs. (9) and (11) that the eigenvalue ω c is positively related to the

length of k + G . From Figure 24, it can be seen that the lengths of k + G0 and k + G1 are

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Dahe Liu, Tianrui Zhai and Zhaona Wang

minimal in both K1 and K2 directions. They correspond respectively to the minimal 2 2 eigenvalue ω c , i.e. the lowest band gaps in two highest symmetric directions. Figure 26 gives the calculated results using Eqs. (10) and (12). It shows that, k + G0 and k + G1 determine the top and the bottom limit of the first gap, respectively. Each direction of

G in Figure 24 corresponds to two band gaps (one for TM wave and the other for TE wave). The top and the bottom edges of the gaps corresponding to TM and TE waves can be obtained by solving Eqs. (9) and (11) when the collection of the values of G and G′ is defined as

{G ,G } (i = 1,2,3,4,5,6) . The value at any point of the band can be calculated by reducing the 0

i

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value of K while keeping its direction unchanged. The inserts in Figure 4 pointed by the arrows are the zoomed figures of the first band gap in K1 and K2 directions respectively, in which the position and the width obtained by PWEM are in good consistency with those obtained by our AS. The bandwidth of TM wave obtained by AS is slightly wider than that obtained by PWEM in K1 direction, while the width of the gaps of TE wave is slightly narrower. In K2 direction, those obtained by the two methods coincide almost perfectly. However, it should be noted that in K1 direction there is no gap for TM wave according to PWEM, but, in AS which is not zero. It indicts that this approximation is more accurate to predict the location of band gaps than the bandwidth of band gaps for TM polarization.

Figure 26. Band gaps calculated by AS and PWEM.

For detail comparison between the two methods, the variation of the top and the bottom band edges with Δn was obtained (see Figure 27) by analyzing Eqs. (10) and (12).

Analytical Solution In Figure 27, it can be seen that: the top and the bottom edges of TM and TE waves in K2 direction are in good superposition. In other words, the dielectric constants can be considered as the same for TE and TM waves in K2 direction when Δn is small. The bandwidth in K1 and K2 directions increase with the increase of the refractive index modulation. The space between the bottom edge in K1 direction and the top edge in K2 direction decreases with the increase of the refractive index modulation, but they won’t intercept when Δn ≤ 0.4 . It means

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Achieving Complete Band Gaps Using Low Refractive Index Material

57

that there is no absolute band gap in low refractive index materials with strict period structure for triangular lattice. But, it shows that the bottom edge in K1 direction and the top edge in K2 direction will superimpose, and absolute band gap will appear when Δn is large enough. Or, in another way of thinking, absolute band gap could be obtained by optimizing the lattice structure, i.e. using a non-strictly periodical structure under low refractive index modulation[28,33].

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Figure 27. Variation of Position of band edges and bandwidth with modulation of refractive index calculated by AS and PWEM.

Plane Wave Expansion Method In Figure 27, it can be seen that: the TM mode degenerates in K1 direction. The top edge of the TE band and the bottom edge of the TM band in K1 direction are in good consistency with those obtained by AS, but, the bandwidth is different from that obtained by AS. While, the band location and the bandwidth of TM and TE waves in K2 direction obtained by the two methods are all well consistent, and show almost no difference when Δn ≤ 0.1 .

(4) Conclusion Analytical solution was obtained for 2D PCs made by holography. It gives concise and intuitionistic physical image as well as the relationship between the characteristics of the band gaps and the parameters of the materials. The results obtained by the analytical solution are in good consistency with those obtained by plane wave expansion method, but, it should be noted that in some directions the solution may not be accurate enough and may be different from the degenerate case in which predicts by PWEM. This analytical approach is practically the first order approximation of the solution of the PC with step distribution of refractive index. Though the analytical solution is derived for 2D triangular lattice made with a special material (DCG), it is valid for any other kind of 2D lattice made by multi-beam interference or

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Dahe Liu, Tianrui Zhai and Zhaona Wang

other gradual refractive index materials. And it must be noted that this solution is accurate only for low refractive index modulation materials.

6. Temperature Tunable Random Lasing in Weakly Scattering Structure Formed by Speckle

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Recently, the low refractive index material DCG was used to implement a weakly scattering disorder structure formed by speckle, and temperature tunable random lasing using this kind of structure was achieved. Since the pioneering work of Ambartsumyan et al. [57], random laser has been a subject of intense theoretical and experimental studies [58-65]. In particular, the discovery of intriguingly narrow spectral emission peaks (spikes) in certain random lasers by Cao et al. [60] has given an enormous boost to such a subject. The random laser represents a nonconventional laser whose feedback is mediated by random fluctuations of the dielectric constant in space. In general, there are two kinds of feedback for the random lasing, intensity feedback and field feedback [64,65]. The former is phase insensitive, which is called as incoherent or nonresonant feedback. The latter is phase sensitive, which can be regarded as coherent or resonant feedback.

Figure 28. (a) Set-up geometry of the disorder structure formed by speckle. H is holographic recording material. (b) and (c) are the microscopic images of the disordered structure formed by speckle.

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The two types of random laser action can not only be found in some strong scattering systems with gain [64-67], they can also be observed in active random media in weakly scattering regime far from Anderson localization. For example, some authors have reported laserlike emission from several weakly scattering samples including conjugated polymer films, semiconductor powders and dye-infiltrated opals [68-72]. In the strongly scattering regime, lasing modes have a nearly one-to-one correspondence with the localized modes of the passive system [64-66,73]. In contrast, the nature of lasing modes in weakly scattering open random systems is still under discussion, although several mechanisms have been proposed to explain such a phenomenon [74-77]. Here we report an experimental observation of random laser in a kind of new random medium in a regime of weak scattering. A disorder structure was formed by speckle. The setup geometry is shown in Figure 28 (a) schematically. A ground glass was used to generate spatial speckle structure. A reference beam was introduced to implement interference for recording the distribution of the spatial speckle in the holographic recording material. Different distribution of the spatial speckle can be recorded by changing the place of the ground glass or the reference beam. The recording material used was dichromated gelatin (DCG) coated on optical glass. The thickness of the DCG used is 36 μm . After exposure, the DCG plate was developed in running water for 120 min at 20 C for developing sufficiently and to remove any residual dichromate, and then soaked into a Rhodamine 6G solution with a concentration of 0.125 mg per milliliter water at the same temperature for 60 min bath enabling the dye molecules to diffuse deep into the emulsion of the gelatin. Then, the DCG plate was dehydrated in turn by soaking it in 50 %, 75 % and 100 % isopropyl alcohol containing the Rhodamine 6G dye with same concentration at 40 C for 15 min. After dehydration, the DCG plate was baked at 100 C for 60 min in an oven. Figure 28 (b) and (c) show the microscopic image of the disordered structure formed by the speckle, and it is a complete disordered structure. The average size of the speckle spot is about 1 μ m . The concentration of the speckle spot is estimated to be the order of magnitude of 109 per cm3. The DCG is a phase type holographic recording material with the refractive index of 1.52 and the modulation of the refractive index less than 0.1. From the transmission measurements, we can estimate the mean free path l * > 80 μ m in the spectral range between 550 nm and 600 nm, which is much larger than the thickness of the system. This means that scattering in the structure is quite weak. We now turn to the investigations on the photoluminescence (PL) of the above dye-doped random samples under a pump field. Figure 29 (a) shows the set-up geometry for the measuring. A nanosecond (ns) pulsed Nd:YAG laser running at 532 nm with repetition rate of 10 Hz, maximum pulse energy of 1500 mJ , pulse width of 8 ns and a picosecond (ps) pulsed Nd:YAG laser running at 532 nm with repetition rate of 10 Hz, maximum pulse energy of 40 mJ , pulse width of 30 ps were used as the pumping sources respectively. The beam diameters of the ns laser and the ps laser were 13 mm and 3 mm respectively. The laser beam was incident on the surface of the optical glass substrate, then penetrates the substrate and entered the dye doped DCG medium. The detector was put close to the sample to collect the energy as much as possible. By rotating the rotation stage the orientation of the speckle structure can be changed with respect to the laser beam. It results in the variation of the wavelength and the peak number of lasing.

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The measured results for the emission spectra under the pump by ps laser are plotted in Figure 29 (b). The blue triangles correspond to the PL spectra of Rhodamine 6G, and the red solid line represents the emission of the dye-doped sample at θ = 25 under the pumping intensity of 120 MW/mm2. The threshold of the random laser is about 50 MW/mm2 as shown in the insets. It can be seen that many lasing modes are excited in the gain spectrum of Rhodamine 6G. The widths of these modes are found to be less than 0.4 nm, which exhibits the coherent feedback very well. The phenomenon is very similar to the previous investigations on the random laser in the other weak scattering systems [68-72]. The origin of the phenomenon can also be understood by the previous theory for such a problem [74-77].

Figure 29. (a) Set-up geometry for measuring random lasing. θ is the angle of the pump beam with respect to the surface of the substrate, it represents the orientation of the sample. (b) Emission pumped by ps laser, and the inset shows the threshold is about 50MW/mm2 at θ = 25° . (c) Emision pumped by ns laser at θ = 30° . In (b) and (c), the blue triangles are the gain spectrum of rhodamine 6G excited by 532 nm laser beam. The black solid line and the red dashed line represent the emissions with pumped energy around and much higher than threshold respectively.

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However, when the dye-doped random sample is pumped by the ns laser, the different phenomenon appears. Figure 29(c) displays the measured results when the pump energy is much higher (320 kW/mm2) than and around (66 kW/mm2) the threshold. A single mode is observed and the profile of the mode is fitted by a Gaussian function, and the result demonstrates that this mode shows a perfect Gaussian profile (as shown in Figure 30(a)). Such a character does not change with the change of the pump energy. When θ = 30° , the emission spectra at different pump energy are shown in Figure 30(b). There is only a single mode at 574 nm. The threshold value is found to be 37.7 kW/mm2, which is much lower than that pumped by ps laser (50 MW/mm2). The lasing becomes more obvious when the pump energy is higher than 65.9 kW/mm2.

Figure 30. (a) Gaussian function fitted emission peak. (b) Measured emission spectra at different pump energy in the case of θ = 30° , and the inset shows the threshold is about 37.7 kW/mm2. The left high spikes （marked by black arrows）correspond to 532 nm pump laser beam, and the appeared width does not show its actual line width of the pump laser because of the saturation effect of the CCD detector used.

In fact, our sample is in the shape of a thin film, it can not be considered as a strict isotropic structure. So, the property of the mode also depends on the pump direction of the laser beam. In some directions, the case with two modes can also be found. Figure 31(a) shows the measured results for the emission spectra of the disordered structure at different pump energy using the ns laser. There are two modes of the emission located at 570 nm and 590 nm respectively and the later is much stronger than the former, and the later becomes the dominant mode when the pump energy is strong enough. When the pump energy is around the threshold value, the two modes are in competition (see Figure 31(b)). In addition, we find that the widths of the modes always keep a few nm for any angle (direction) and intensity of the ns pump beam, which displays the character of intensity feedback. In general, the intensity feedback corresponds to diffusion motion of photons in

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Dahe Liu, Tianrui Zhai and Zhaona Wang

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active random medium, when the photon mean free path is much smaller than the dimension of scattering medium but much longer than the optical wavelength [64,65]. It can be described theoretically by the diffusion equation for the photon energy density in the presence of a uniform and linear gain [57,64,65]. Obviously, it is not such a case for the weak scattering system in the present work, because the mean free path is much larger than the thickness of the sample. However, our experimental results demonstrate that incoherent feedback can also be realized in such a non-diffusion random system by choosing suitable pump sources. In the previous investigations, one has observed the transition between coherent feedback and incoherent feedback by varying the amount of scattering in the gain medium [64,78]. In fact, our present work illustrates that such a transition can also be realized in the same random sample by changing the pump source.

Figure 31. (a) Measured radiation spectra at different pump energy in the case of θ = 18° . (b) Competition of two modes around the threshold value in the case of θ = 18° . The left high spikes (marked by black arrows) correspond to 532 nm pump laser beam, and the appeared width does not show its actual line width of the pump laser because of the saturation effect of the CCD detector used

The phenomena can be understood in terms of the following analysis. For the case of ns laser pumping, the light will go through a path about several meters in the period of one pulse. It means that the light will have much more times of scattering, i.e. has much longer effective interaction length. Therefore, the mode competition has finished and one mode becomes dominant. That is the reason only a single mode is dominant in the emission as long as the pumping energy is higher. However, for the case of ps laser pumping, the light will go through a path about several millimeters. It means that the effective interaction length is much

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63

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shorter, so the mode competition can not be finished even though the pumping energy is rather high. Besides, for the case pumped by ps laser, because the effective interaction length is very short, i.e., the scattering times is much less. Therefore, much higher pump energy is needed for building up the lasing, it results in the high threshold. Another interesting phenomenon for the present system is that the emission wavelength for the random laser can be tuned through changing temperature. Figure 32 shows the measured spectra of lasing emission at different temperatures. It can be seen clearly that the emission peaks shift toward to the long wavelength with the increase of the temperature. The lasing will be kept so long as the frequency of the emission is still inside the gain profile when temperature changes. In fact, the temperature tuning for the random laser by infiltrating sintered glass with laser dye dissolved in a liquid crystal had been investigated in previous work [62]. The diffusive feedback was controlled through a change of refractive index of the liquid crystal with temperature. However, such a tuning was employed to turn on and off random lasers. In contrast, the wavelength tuning can be realized in the present system. The phenomenon originates from the special property of the material. With the changes of the temperature, the light path changes due to variation of the distance between the two adjacent speckle spots. It makes the variation of the effective length of interaction, and then the lasing wavelength is changed.

Figure 32. Tunability of wavelength vs. temperature of the random laser.

Finally, it should be pointed out that, the DCG can be coated on a soft substrate, such as polyethylene terephthalate (PET). So, the speckle structure recorded by holography can be formed as a soft film. On the other hand, the holography recorded speckle structure is pretty stable. Hence, the output of this kind of laser has good stability and repeatability. It will be beneficial for actual applications. We did implement the speckle random laser using DCG coated on the PET soft substrate, and the same results mentioned above were obtained. In summary, we have fabricated a weakly scattering disordered structure formed by speckle using holography. In such a system with gain, we have observed low-threshold random lasing with two kinds of feedback, incoherent and coherent, for the same sample in the case pumped by nanosecond and picosecond lasers, respectively. The wavelength tunability of the random laser with the change of the temperature has been demonstrated. Our results can not only get a deeper understanding on the open question about the properties of

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Dahe Liu, Tianrui Zhai and Zhaona Wang

the lasing modes in weakly scattering regime, it can also benefit some applications such as remote temperature sensing in hostile environments.

References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10]

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[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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[52] Jakubiak, R; Tondiglia, VP; Natarajan, LV; Sutherland, RL; Lloyd, P; Bunning, TJ; Vaia, RA. “Dynamic lasing from all-organic two-dimensional photonic crystals,” Adv. Mater., 2005, vol. 17, 2807-2811. [53] Zheng, J; Ye, Z; Wang, X; Liu, D. “Analytical solution for band-gap structures in photonic crystal with sinusoidal period,” Phys. Lett. A, 2004, vol. 321, 120-126. [54] Samokhvalova, K; Chen, C; Qian, B. “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys., 2006, vol. 99, 063104. [55] Nusinsky, I; Hardy, A. “Approximate analysis of two-dimensional photonic crystals with rectangular geometry. I. E polarization,” J. Opt. Soc. Am. B, 2008, vol. 25, 11351143. [56] Joannopoulos, JD; Johnson, SG; Winn, JN; Meade, RD. Photonic Crystals: Molding the Flow of Light, 2nd ed. Princeton, NJ:Princeton Univ. Press, 2008, 49-52. [57] Ambartsumyan, RV; Basov, NG; Kryukov, PG; Letokhov, VS. IEEE J. Quantum Electron. QE-2, 442 (1966), V. S. Letokhov, Sov. Phys. JETP, 26, 835, 1968. [58] Lawandy, N; Balachandran, R; Gomes, A; Sauvain, E. Nature (London), 368, 436 1994. [59] Wiersma, D; van Albada, M; Lagendijk, A. Phys. Rev. Lett, 1995, 75, 1739. [60] Cao, H; Zhao, YG; Ong, HC; Ho, ST; Dai, JY; Wu, JY; Chang, RPH. Appl. Phys. Lett, 73, 3656, 1998, Phys. Rev. Lett, 1999, 82, 2278. [61] Jiang, XY; Soukoulis, CM. Phys. Rev. Lett, 2000, 85, 70. [62] Wiersma, D; Cavalieri, S. Nature (London), 2001, 414, 708-709. A. Rose, Zhengguo Zhu, Conor F. Madigan, Timothy M. Swager and Vladimir Bulovic, Nature (London), 2005, 434, 876. [63] Hanken, E. Türeci, Li Ge, Stefan Rotter, A.Douglas Stone, Science, 2008, 320, 643. [64] Cao, H. Waves Random Media, 2003, 13, R1 2003. [65] Wiersma, DS. Nature Physics, 2008, 4, 359. [66] Milner, V; Genack, AZ. Phys. Rev. Lett, 2005, 94, 073901. [67] Fallert, J; Dietz, R; Sartor, J; Schneider, D; Klingshirn, C; Kalt, H. Nature Photon, 2009, 3, 279. [68] Frolov, SV; Vardeny, ZV; Yoshino, K; Zakhidov, A; Baughman, RH. Phys. Rev., B, 1999, 59, R5284. [69] Ling, Y; Cao, H; Burin, AL; Ratner, MA; Liu, X; Chang, RPH. Phys. Rev. A, 2001, 64, 063808. [70] Mujumdar, S; Ricci, M; Torre, R; Wiersma, D. Phys. Rev. Lett, 2004, 93, 053903. [71] Polson, RC; Vardeny, ZV. Phys. Rev. B, 2005, 71, 045205. [72] Wu, X; Fang, W; Yamilov, A; Chabanov, A; Asatryan, A; Botten, L; Cao, H. Phys. Rev. A, 2006, 74, 53812. [73] Jiang, X; Soukoulis, CM. Phys. Rev. E, 2002, 65, 025601. [74] Apalkov, VM; Raikh, ME; Shapiro, B. Phys. Rev. Lett, 2002, 89, 016802. [75] Florescu, L; John, S. Phys. Rev. Lett, 2004, 93, 13602. [76] Deych, L. Phys. Rev. Lett, 2005, 95, 043902. [77] Vanneste, C; Sebbah, P; Cao, H. Phys. Rev. Lett, 2007, 98, 143902. [78] Cao, H; Xu, JY; Chang, SH; Ho, ST. Phys. Rev. E, 2000, 61 1985.

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In: Photonic Crystals Editor: Venla E. Laine, pp. 67-93

ISBN: 978-1-61668-953-7 © 2010 Nova Science Publishers, Inc.

Chapter 3

PHOTONIC CRYSTALS FOR MICROWAVE APPLICATIONS Yoshihiro Kokubo Graduate School of Engineering, University of Hyogo Himeji-shi, Hyogo Japan.

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Abstract This chapter introduces photonic crystals for microwave, millimeter wave and submillimeter wave applications. First, we present a new type of photonic crystal waveguide for millimeter or submillimeter waves which consists of arrayed hollow dielectric rods. The attenuation constant of the waveguide is smaller than that of a Nonradiative Dielectric (NRD) Waveguide. Second, metallic photonic crystals consisting of metamaterials are designed. Metamaterials are artificially-constructed materials whose permittivity and/or permeability can be negative. A hollow metal rod with a slit in its sidewall is a metamaterial which plays the role of an LC resonator. In addition, an array of such metal rods can behave as a twodimensional photonic crystal. Conventional metallic photonic crystals have a photonic bandgap for electromagnetic waves with E-polarization (where the electric field is parallel to the axis of the rods). On the other hand, metamaterials consisting of hollow metal rods with sidewall slits have a bandgap for electromagnetic waves with H-polarization (where the electric field is perpendicular to the axis of the rods) because of their negative permeability. Eigenvalues can be calculated using the two-dimensional finite-difference time-domain (FDTD) method, and metallic metamaterial-based photonic crystals can be designed with independently customized bandgaps for E- and H-polarizations. Third, we analyze propagation losses in metallic photonic crystal waveguides. A conventional metallic waveguide structure whose E-planes are replaced by metal wires can easily confine electromagnetic waves because their electric field direction is parallel to the wires. In this section, the attenuation constants of metallic photonic crystal waveguides, whose E-planes are replaced by arrayed metal rods, are calculated and compared with those of conventional metallic waveguides. Losses are calculated using another FDTD method. The propagation losses are found to decrease when the radii of the rods (relative to the lattice constant) is increased. The bandwidth can be extended by widening the waveguide relative to the lattice constant. Fourth, we investigate a metallic waveguide in which in-line dielectric rods are inserted. In such a situation, the electromagnetic field is equivalent to that produced by a twodimensional photonic crystal with a lattice constant in one direction equal to the inter-rod spacing, and in the other direction equal to twice the distance between the rods and the

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Yoshihiro Kokubo sidewalls. The calculations of the fields are performed using the FDTD method. Our results showed that the frequency range of a single propagation mode in the waveguide can be extended. Finally, arrayed dielectric rods in a metallic waveguide are known to change the modes and of course the group velocities of electromagnetic waves. In particular, we focus on TE10 to TEn0 (where n equals 2 or 3) mode converters and the dependence of their behavior on the frequency range. Conventional metallic waveguides only allow propagation at frequencies above the cutoff frequency fc. The TE20 mode, however, is also possible for frequencies higher than 2 fc. In this investigation, a mode converter is proposed which passes the TE10 mode for frequencies less than twice the cutoff frequency, and converts the TE10 to the TE20 mode for frequencies higher than twice the cutoff frequency; this is achieved by small variations of the group velocity.

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1. Introduction Photonic crystals (PCs) are periodic optical structures that are designed to control the motion of photons. PCs are generally designed to be used at optical frequencies. However, they also have the potential to be used at microwave or millimeter-wave frequencies. There are currently no satisfactory waveguides for millimeter- and submillimeter-wave frequencies, because conventional waveguides suffer from dielectric and metallic losses. The few waveguides used at these frequencies suffer from high propagation losses. Nonradiative dielectric (NRD) waveguides [1] are used as waveguides for millimeter-wave frequencies and they do not have bending losses; however, their losses in a straight line are high. A waveguide is required with low propagation losses in these frequency ranges. PCs consisting of two-dimensional (2D) square lattices of rods have no photonic bandgaps for H-polarization (when the magnetic field is parallel to the rods) unless the rod radius is sufficiently high. Another problem is that the characteristics of conventional photonic crystals change for E-polarization (when the electric field is parallel to the rods) when the characteristics for H-polarization are changed. It would be convenient if the bandgap of a PC for H-polarization is independent of that for E-polarization. Metamaterials are artificial materials that have characteristics that differ from those of natural-occurring materials and which can be controlled by varying the configurations of conventional materials. Metamaterials have negative permeabilities and so electromagnetic waves cannot propagate in them. Using metamaterials, it may be possible to produce PCs that have novel characteristics. A metallic waveguide structure whose E-planes are replaced by metal wires can easily confine electromagnetic waves because the electric field direction is parallel to the wires. In this section, the attenuation constants of metallic photonic crystal (MPC) waveguides whose E-planes are replaced with metallic rod arrays are calculated and compared with those of conventional metallic waveguides. Metallic waveguides have several major advantages, including low propagation losses and high power transmissions at microwave frequencies. However, they have the disadvantage that the propagation frequency band is limited above the cutoff frequency, fc. The usable frequency range is restricted to fc < f < 2 fc since the TE20 mode can exist only at frequencies higher than 2 fc in rectangular metallic waveguides. A ridge waveguide [2] has a larger propagating frequency range because it has a lower cutoff frequency for the TE10 mode. However, it has the disadvantage of a high attenuation constant. PCs have bandgaps in

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which electromagnetic waves cannot propagate. This characteristic can prevent the TE10 mode propagating and permit only the TE20 mode to propagate at frequencies higher than 2 fc in a metallic waveguide [3]. Power dividers and power combiners may be easily setup using mode converters. For example, a TE10–TE30 mode converter easily offers a three-port power divider, and a threeway power combiner can be composed by reversal. A power combiner is useful for application to Gunn diodes in a waveguide array, because it converts the TE30 mode to the TE10 mode. We have already proposed a mode converter that allows the TE10 mode to propagate at low frequencies and efficiently converts the TE10 mode into the TE20 mode at high frequencies [4]. In this section, a new mode converter is proposed which passes through the TE10 mode for the low frequency range and efficiently converts TE30 to TE10 mode for the high frequency range.

2. A New Type of Photonic Crystal Waveguide for Millimeter-Wave Frequencies

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Photonic crystals (PCs), which are photonic bandgap materials, are theoretically predicted to confine and guide light along a channel without bending loss [5]. Waveguides made from PCs are useful optical waveguides since propagation losses may be neglected at optical frequencies due to the small sizes of silicon wafers. However, PC waveguides will have high losses at millimeter-wave frequencies if the same structures are used as those at optical frequencies. We have investigated PC waveguides that have a triangular lattice of hollow dielectric posts and that have lower losses than NRD waveguides at millimeter-wave frequencies.

2.1. Structure of PC Waveguide We first need to design a PC structure that can reflect electromagnetic waves. A twodimensional PC structure consisting of a triangular lattice with aligned hollow dielectric posts is suitable for waveguides because it is easier to fabricate a waveguide system using such a structure than with the conventional structure of a dielectric slab containing many holes. Calculations based on the plane wave expansion method predict that the dielectric constant εr needs to be greater than 15 for there to be a bandgap in a triangular lattice of hollow dielectric posts for both H-polarization (where the electric field is perpendicular to the posts) and Epolarization (where the electric field is parallel to the posts) at the same frequency when ωa/2πc < 1 and the mode number ≤ 10 [6]. In this section, we assume that εr = 24 (for example, LaAlO3, tan δ = 3×10−4 at 10 GHz). Figure 1 shows the bandgap map of a triangular lattice with hollow dielectric posts. The external radius r1 was fixed to a/2 (where a is the lattice constant) to ensure that the posts are in contact with each other. The propagation mode in the PC waveguide must be a hybrid mode that has field components in all directions. Since hybrid modes cannot be separated into H- and Epolarizations, a bandgap for either H- or E-polarization prevents electromagnetic waves from propagating in the PC. If we fix the normalized internal radius r2/a to 0.4, the first bandgap lies between 0.23 < ωa/2πc < 0.37 for H-polarization.

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Yoshihiro Kokubo

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Figure 1. The bandgap map of a triangular lattice with hollow dielectric posts. a represents the lattice constant and r2 is the internal radius (external radius r1 = a/2)

Figure 2. The structure of PC waveguide for 50 GHz. One row of posts has been replaced with a dielectric slab

Figure 3. Electric fields in a dielectric slab guide and a PC waveguide. The spreading width is defined as the distance y from the center of the slab to the point at which the intensity of the electric field is one tenth that of the peak

We need to determine the structure of the waveguide for millimeter-wave frequencies after the structure of the PC has been fixed. Of the many structures consisting of dielectric slabs, PCs and metals that we have investigated, the best structure appears to be a slab Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

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between PCs. The primary electric field in the slab is in the x direction and only a dominant mode can propagate. An electromagnetic wave at the center of the slab decays exponentially in the y direction. This spread in the electromagnetic wave depends on the slab width D, making it difficult to determine the length of the posts. We compare the spread of the electric field in the PC waveguide and in a conventional dielectric slab guide when the electric field is parallel to the slab. Figure 3 shows the distance from the center at which the intensity of the electric field is one tenth that of the peak at 50 GHz. The spread in the electric field in the dielectric slab guides is around 4 mm when the slabs are 1.5 mm to 3 mm thick, but it increases drastically when the thickness is less than 1 mm. The electric field in the PC waveguide spreads in a similar manner but it is slightly wider than that in the dielectric slab guide. Since the length of the posts needs to be three to four times longer than the spreading width, we fixed the length of the posts to 28 mm for D greater than 1 mm.

2.2. Propagation Loss 2.2.1. Dielectric and Metallic Losses Generally, the attenuation constant of a waveguide is given by

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α = (Pl,D+Pl,M)/2P

(1)

if the losses are small. Pl,D and Pl,M are the dielectric and metallic losses, respectively. P represents the travelling wave power. The minimum propagation losses of an NRD waveguide would be 1.7 dB/m (f = 30 GHz), 3.3 dB/m (f = 50 GHz) and 7.8 dB/m (f = 100 GHz) when the dielectric constant of the slab εr = 2 and tan δ = 2×10−4 when the higher LSM and LSE modes are cutoff. P, Pl,D and Pl,M are given by

P=

1 1 ( E × H *) z dS = ∫∫ ( E x H *y − E y H x* )dxdy ∫ 2S 2 x, y

1 2 Pl , D = ω ∫ ε ⋅ tan δ ⋅ E dV 2 V

(

2 1 2 = ωε 0 ∫ ε r ⋅ tan δ ⋅ E x + E y + E z 2 V

Pl , M =

1

H 2δ σ ∫ c

=

⎛ ⎜ Hy ⎜ 2δ cσ ⎜ y , z ∫∫ ⎝ at side walls 1

(

2

+ Hz

2

2 t

2

(2)

)⋅ dxdydz

(3)

dS

S

)dydz+

∫∫ ( H

⎞

2

x x, z at top and bottom walls

+ Hz

2

) dxdz ⎟⎟ ⎟ ⎠

where σ represents the conductivity of the guided walls and δc is their skin depth. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

(4)

72

Yoshihiro Kokubo

Losses of PC waveguides are calculated using the three-dimensional FDTD method [7]. In this case, P and Pl,D are given by

P=

w h 1 Δx ⋅ Δy ⋅ ∑∑ (E x ( x, y, z ) H y ( x, y, z ) 2 x =0 y =0

− E y ( x, y , z ) H x ( x , y , z ) )

w h unitlength 1 P1, D = ωε 0 ⋅ Δx ⋅ Δy ⋅ Δz ⋅ ∑∑ ∑ ε r tan δ × 2 x =0 y =0 z =0

(E

2 x

( x , y , z ) + E y ( x, y , z ) + E z ( x, y , z ) 2

2

(5)

(6)

)

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where w is the waveguide width (x direction), h is its height (y direction), and E(x,y,z) and H(x,y,z) are the maximum values of the electric and magnetic fields over time at each point, respectively.

Figure 4. Calculation results for propagation loss.

Because electromagnetic waves in a periodic structure essentially generate reflected waves, it may be difficult to directly determine the net travelling wave power using Eq. (5). However, since the time-averaged power flow must be same at any cross section in the waveguide in the steady state, the net travelling wave power P is given by

P = Δx ⋅ Δ y ⋅

1 N

∑∑∑ (E T

w

h

t =0 x =0 y =0

x

( x, y , z , t ) H y ( x, y , z , t )

− E y ( x , y , z , t ) H x ( x, y , z , t ) )

Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

(7)

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where E(x,y,z,t) and H(x,y,z,t) represent the intensities of the electric and magnetic fields at each time respectively, and T is the period for one cycle and N is the time step of one cycle.

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Figure 5. Attenuation constant versus slab width D.

Figure 4 shows the attenuation constants of the PC waveguide around 50 GHz. Dielectric losses in the PC layers are relatively large compared with those in the dielectric slab and have minimum values near 46-47 GHz and 56-57 GHz. Because electromagnetic waves do not penetrate deeply into the PC at frequencies near the middle of the bandgap, the dielectric loss generally tends to be small. A bandgap exists between 38.3 GHz and 61.8 GHz for Hpolarization and between 55.1 GHz and 58.1 GHz for E-polarization. Consequently, there are two minimum propagation losses in the PC waveguide. Attenuation constants for various slab widths D at 50 GHz are calculated and shown in Figure 5 since they depend on the expanse of the propagating wave. As expected, the attenuation constants for both regions are enhanced when D is increased.

2.2.2. Bending Loss We also calculate the bending loss of the PC waveguide because many studies have found that a conventional PC waveguide suffers from reflection at bending corners. According to calculations using the FDTD method, the bending loss is 1−1.5 dB at a 60degree bend at frequencies between 45−58 GHz where a bandgap exists for H- and Epolarizations (see Figure 4). The difference between the bending loss and the power reflection may due to scattering because of the limited length of the posts. The length of the posts is determined from calculation results for a straight guide. Therefore, we can calculate the electric field at a bending corner to determine whether the posts are sufficiently long. Figure 6 shows the electric field Ex at x = 0 (center of the slab). Ex decays exponentially to the outside of the slab in straight sections at some times. On the other hand, Ex at a bending corner tends to decrease due to vibrations to the outside of the slab. We set the length of the posts to 28 mm, this may be long enough at a bending corner.

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Figure 6. Comparison of electric field (Ex) distributions in a straight slab and at a bending corner (D = 0.90 mm).

Figure 7. Attenuation constants of a PC waveguide and an NRD waveguide versus frequency.

Figure 8. Gap map of arrayed metal column arranged in two dimensional square lattice.

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2.2.3. Comparison of Loss Frequency Characteristics of Propagation and NRD Waveguide Figure 7 shows the minimum attenuation constants of an NRD waveguide at millimeterwave frequency when εr = 2, tan δ = 2×10−4 and each dimension is varied with frequency when higher LSM modes cutoff. The attenuation constants increase superlinearly because of metallic losses. Figure 7 also shows the attenuation constants of a PC waveguide given by Eq. (3). If it is assumed that εr = 2, tan δ = 2×10−4 and D = 0.165λ for the slab and a = 0.33 λ for the PC waveguide, the attenuation constants will be 1.1 dB/m (f = 30 GHz, D = 1.65 mm, a = 3.3 mm), 2.1 dB/m (f = 55 GHz, D = 0.9 mm, a = 1.8 mm) and 3.8 dB/m (f = 100 GHz, D = 0.495 mm, a = 0.99 mm). The PC waveguides have lower losses than NRD waveguides in microwave and millimeter-wave frequency ranges.

3. Two-Dimensional Photonic Crystals Using Metamaterials Photonic crystals arranged in two-dimensional (2D) square lattices have no photonic bandgaps for H-polarization unless they have sufficient column radii as shown in Figure 8. In addition, the characteristics of conventional photonic crystals are changed for E-polarization when the characteristics for H-polarizations are changed.

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3.1. Metamaterials Metamaterials are artificial materials with characteristics that are not available in natural materials and which can be controlled by changing the configurations of conventional materials. One of the metamaterials is the split-ring column structure shown in Figures 9 and 10, which has radius r, lattice constant a, ring thickness t, and ring gap dg. After application of a magnetic field H0 along the axis of the column, resonation occurs from the current in the ring and ring gap. The effective permeability μ, for example, is varied with respect to the frequency, shown in Figure 11. If the frequency dependence of μ follows the Drude model, then a bandgap is generated at μ cπ/a. If the eigenvalues of the TE10 mode turn back at k = π/a, then single mode propagation is available for the TE20 mode above 2fc. If we let f be 2 fc in ω (= 2πf) = cπ/a, then this means that the width of the waveguide, w, is set to λc/2 = (c/fc)/2 = 2a. The periodic structure is assumed to be due to the dielectric rods placed at the center of the waveguide, because this will have a strong influence on the TE10 mode and will not affect the TE20 mode.

5.2. Waveguide Structure

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A detailed profile of the wide-band waveguide is shown in Figure 23. An array of circular dielectric rods is set to the center of a metallic waveguide. The dielectric material is assumed to be easily obtainable LaAlO3 (lanthanum aluminate) with a dielectric constant, εr, of 24 and a loss tangent, tanδ, of 3×10-4 (at 10 GHz). If the rectangular waveguide is WR-90 (22.9×10.2 mm； fc ≈ 6.55 GHz), a is determined as w/2 = 22.9 mm/2 = 11.45 mm. The propagation modes for this waveguide are calculated using the supercell approach [16], by applying appropriate periodic Bloch boundary conditions for the unit cell [17]. The supercell profile is also shown in Figure 23. w is set to w = λc/2 = c/fc/2 = 2.4a and a = 9.54 mm. The results calculated for r = 0.054a are shown Figure 24 (r represents the radius of the dielectric rod). In this figure, the first band has even symmetry and the second band has odd symmetry, and they are both similar to the TE10 and TE20 modes in a rectangular metallic waveguide, respectively. The first band is from 6.0 to 13.2 GHz and the second band becomes wider, from 13.2 to 16.8 GHz, and from 17.6 to 21.4 GHz, because the third band is elevated and flattened. The normalized bandwidth of this waveguide WB = (Δω1+Δω2+Δω3)/ω0 ≈ 0.95.

Figure 23. Detailed profile of a WR-90 waveguide in which the dielectric rods are aligned. [3]

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Figure 24. Dispersion of the eigenvalues along the guide axis for εr=24, r/a =0.054, and w/a = 2.4. (The right-hand vertical axis shows the single-mode frequency when a is 9.54 mm.) [18]

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5.3. Structure of a 90-Degree H-Plane Bent Waveguide [18] When a TE10 mode is introduced between 6.0 and 13.2 GHz and a TE20 mode at 13.2– 16.8 or 17.6–21.4 GHz, single-mode propagation is possible. Dielectric rods are only required at the bent portions because mode conversion does not occur in the straight segments. Reflection, however, will occur at the borders if the dielectric rods are removed along the straight portions. To reduce those reflections, the radius of the rods at the boundaries can be made small. The 90-degree bent waveguide structure of Ref [23] contains an integral number of dielectric rods. Consequently, rods are located at the corners of the bent portions where they begin or end. In this case, the curvature radius cannot be chosen arbitrarily and one cannot use commercial bent waveguides.

Figure 25. 90-degree H-plane bent metallic waveguide.

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On the other hand, Figure 24 shows a case in which the radius of curvature R and period a are fixed. Dielectric rods are not necessarily located at the corners of the 90-degree bent portion and three rods are included in the straight portions. The last dielectric rod is thin with r = 0.36 mm in order to minimize reflections of the TE10-like mode. The S parameters of the 90-degree H-plane bent waveguide for R = 40 and 50 mm are plotted as solid lines in Figure 26 using HFSS (high frequency structural simulator) software by Ansoft [24]. These different curvatures (R = 40 mm and R = 50 mm) give similar results. The frequency range is from 8.2 to 10.4 GHz for the first band and from 13.5 to 16.7 and 18.4 to 19.2 GHz for the second band with reflection below –20 dB. The TE10 mode transmittances are also shown at the second band. The mode conversions are very small. The TE30 cutoff is less than 19.6 GHz.

(a)

(b) Figure 26. Transmission coefficient |S21| and reflection coefficient |S11| at the corner of a waveguide for (a) R = 40 mm and (b) R = 50 mm. [18]

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Figure 27 (a) Dielectric rod located in a waveguide that does not have gaps at the top and bottom. (b) Dielectric rod inserted in a hole drilled at the top of the waveguide with a diameter of 0.2 mm that is larger than the diameter of the rod. [18]

5.4. Simple Fabrication Method [18] It is not trivial to position dielectric rods in a waveguide such as Type A illustrated in Figure 27(a). One must locate the dielectric rods in the waveguide which has no gap at its top or bottom. As a solution, 0.2-mm-diameter holes were drilled into the top of the waveguide and dielectric rods were inserted through them, resulting in Type B in Figure 27(b). The S parameters were calculated using the HFSS software and the results are graphed as dotted lines in Figures. 26(a) and (b). The results for these different structural conditions (solid lines and dotted lines) are similar. The difference between the S parameters of Type A and B is even smaller if the drilled holes are narrower than 0.2 mm.

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6. Frequency Range Dependent TE30 to TE10 Mode Converter A structure in which two arrays of dielectric rods are setup can convert TE30 to TE10 mode [25]. We have reported a mode converter which passes through a TE10 mode for low frequency range and converts TE10 to TE20 mode for the high frequency range [4].

Figure 28. The frequency eigenvalues of a conventional metallic waveguide in a given k wavevector.

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Power dividers and power combiners may be easily setup using mode converters. For example, a TE10–TE30 mode converter easily offers a three-port power divider, and a threeway power combiner can be composed by reversal. A power combiner is useful for application to Gunn diodes in a waveguide array, because it converts the TE30 mode to the TE10 mode. In this investigation, a new mode converter is proposed which passes through the TE10 mode for the low frequency range and efficiently converts TE30 to TE10 mode for the high frequency range. The frequency eigenvalues of a conventional metallic waveguide in a given k wavevector are shown in Figure 28. In this figure, the wavevector k and frequency ω are normalized using the width of the waveguide w. The electromagnetic wave propagates the TE10 mode only for 0.5 0 and does not evolve from Fabry-Perot modes resonantly excited within the linear limit of the barrier media. In Table 1 a listing of λ|En,0 |2 for n = 6, 4, 2, 0, -2, -4, -6 in the seven Kerr sites of the barrier is given for some of the points in Fig. 3 that are intrinsic localized modes resonances. The wave function intensities are highly peaked at the center of the barrier media and decay rapidly to small values at the edges of the barrier. This is a characteristic of intrinsic localized modes whose large peak intensities modify the Kerr dielectric constant of the barrier media so as to maintain and support the localized mode. It is a self-consistent arrangement with the mode modifying the dielectric which in turn allows the mode to exist within the dielectric. The most highly peaked intrinsic localized modes occur at low values

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of g, while with increasing g the peak heights of the intrinsic localized modes decrease. Eventually the ridge of intrinsic localized mode solutions rises in (r, g) space to meet the lowest branch of Fabry-Perot modes. As r is increased further, regions of multiple intrinsic localized modes (for 0.8 < g < 1.0 and r > 1.0) and dark soliton modes (off the r scale of the figure) are found. These have been discussed by us elsewhere[39, 40, 42, 53]. In this Chapter the focus will be on intrinsic localized modes and their properties in the simple barrier and more complicated barriers considered later.

1.2

g

1 ILM 0.8

0.6 0

0.5

1

1.5

2

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r Figure 3. Plot in the space of (r, g) of the locations of the resonant transmission peaks from the simple barrier of seven Kerr sites shown in Fig. 1b. The Fabry-Perot modes are contained within lines beginning at r = 0 and continuing across the r > 0 plot. The intrinsic localized modes start at r 6= 0 at the lowest values of g and, as r increases, rise to meet the lowest line of Fabry-Perot modes. Next to the lower right of the region of intrinsic localized modes the notation ILM has been added in the plot. The Fabry-Perot modes corresponding to different lines in the plot have different types of electromagnetic intensity profiles within the barrier media while those corresponding to the same line have similar electromagnetic intensity profiles within the barrier media. The intrinsic localized modes, of the type discussed above, show a large center peak intensity created by a small incident wave field intensity[45, 46]. In device applications this offers for a way of providing for the creation of a large field intensity within a dielectric medium through the modulation of a low intensity guided wave, i.e, it offers an amplification function for the incident modulated field. This is similar to the amplification of of an electronic signal provided by certain types of electronic circuits. In addition, the conditions on the system for the excitation of such modes are restrictive so that small variations in system parameters allow for a switching effect. For example, varying the intensity of the incident guided wave can substantially change the amount of the incident wave transmitted or reflected by the barrier. This affect may be elaborated upon, and for an intensity modulated guided mode the Kerr barrier can also act as a type of half-wave rectifier. This is done by off-tuning the average field intensity of the guided mode from the transmission peak of a barrier intrinsic localized mode so that only during part of the modulation cycle would the intrinsic localized mode be resonantly excited with an associated resonant transmission of the guided mode.

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               ÆÆÆÆÆÆÆÆÆÆ ÆÆÆÆÆÆÆÆÆ   Æ 

         Æ  Æ  Æ  Æ  Æ  Æ  Æ ÆÆÆÆ Æ Æ Æ Æ Æ Æ ÆÆ 

Figure 4. Schematic drawing of: a) a barrier coupled at its center site to an off-channel linear or Kerr nonlinear impurity site and b) a barrier containing a linear or Kerr nonlinear impurity site at its center. In the drawing only the waveguide and impurity sites are shown and the sites of the bulk photonic crystal are suppressed. The open circles are linear media waveguide sites, the open circles with the small dots in the waveguide channel are Kerr nonlinear media sites, and the off-channel or center barrier site open circle with the larger dot is a linear or Kerr nonlinear media impurity. The lattice constant of the waveguide is the same as that of the bulk photonic crystal. For simplicity the closed dark circle sites of the bulk photonic crystal are omitted.

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g

1.2

1 ILM

0.8 0

0.5

1

1.5

2

r Figure 5. Plot in the space of (r, g) of the locations of the resonant transmission peaks from a simple barrier of seven Kerr sites coupled at the center nonlinear site to an off-channel site of linear dielectric media described by gi = 5. Aside from the off-channel site, the system is the same as in Fig. 3. The intrinsic localized modes present in Fig. 3 are seen to be split by the off-channel coupling. The notation ILM has been added at the lower right of the region of the intrinsic localized modes.

2.2.

Kerr Media Barrier with Linear Dielectric Off-Channel Feature

A generalization of the Kerr media barrier problem is to couple the center site of the barrier to an off-channel impurity site formed by cylinder replacement of a photonic crystal site positioned next to the center barrier site and off the waveguide channel. A schematic Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

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Table 1. Intrinsic localized mode field intensities λ|En,0 |2 on the simple barrier sites for n = 6, 4, 2, 0,-2, -4,-6 for selected values of (r, g) and for k = 2.9.

(r, g) (0.3, 0.9433) (0.4, 0.9570)

n=6 0.0002 0.0003

Intrinsic Localized Modes n = 4 n = 2 n = 0 n = −2 0.0026 0.0234 0.1197 0.0356 0.0038 0.0260 0.0948 0.0261

n = −4 0.0027 0.0039

n = −6 0.0002 0.0003

drawing of this case is given in Fig. 4a, and we shall take the off-channel impurity site to be of linear dielectric media. The equations for this geometry are the same as those in Eqs. (1) through (5) with the modifications that now the equation E0,0 = g(1 + λ|E0,0 |2 )E0,0 + gl b(E−1,0 + E1,0 )

(8)

from Eq. (2) is replaced by E0,0 = g(1 + λ|E0.0 |2 )E0,0 + gl b(E−1,0 + E1,0 ) + gi bE0,−1

(9)

and an equation for the off-channel site given by

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E0,−1 = gi E0,−1 + gb(1 + λ|E0,0 |2 )E0,0

(10)

is added to complete the system of equations for the case described by Fig. 4a. In Eqs. (9) and (10) gi is related to the dielectric media of the off-channel replacement cylinder and may differ from the gl of the replacement cylinders forming the waveguide channel. In the limit that gi → 0 the off-channel replacement cylinder reduces to a cylinder of the bulk photonic crystal so that the equations of the barrier with an off-channel site reduce to those of the simple barrier shown in Fig. 1b. The modified Eqs. (1) through (5) and Eqs. (8) through (10) are solved for the same incident, reflected, and transmitted wave boundary conditions given in Eqs. (6) and (7), using the same values of b, λx2 and wavenumber k from which the results in Figs. 2 and 3 were obtained. In Fig. 5 the resulting plot of locations of the resonant transmission peaks in (r, g) space is shown for the case in which gi = 5.0. Only peaks for which T > 0.6 are indicated in the plot. As in Fig. 3 a series of seven lines of Fabry-Perot modes are observed. These lines begin at r = 0 for g > 1.0, and the six top lines continue across the entire plot while the lowest line of Fabry-Perot modes ends near r = 0.7. In addition, a series of lines of intrinsic localized modes is also found in the plot. These occur for g < 1.0 and begin at r > 0. The notation ”ILM” is added to the lower right of the series of lines of intrinsic localized mode solutions occurring in the region of (r, g) for 0.2 < r < 1.0 and 0.9 < g < 1.0. It is seen that the off-channel impurity splits the original branch of intrinsic localized modes of the simple barrier in Fig. 3 into three branches of intrinsic localized modes. Again the branches of intrinsic localized modes are in the lower left corner of the plot, with the branches appearing at values of r > 0.20. As r increases, one of the branches rises in g to meet the lowest valued branch in g of Fabry-Perot modes. The other two branches are of a more horizontal nature as r increases, and both cross the rising branch

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Table 2. Listing for the barrier with coupling to an off-channel linear media site of the intrinsic localized mode field intensities. Valves are shown for λ|En,0 |2 in the barrier sites for n = 6, 4, 2, 0, -2, -4, -6 for selected values of (r, g) and gi for k = 2.9 and λx2 = 0.001. (r, g) (0.4, 0.9519) (0.4, 0.9593) (0.6, 0.9682) (0.6, 0.9897) (r, g) (0.6, 0.9711) (0.6, 0.9739) (0.6, 0.9846)

Intrinsic Localized Modes for gi = 5.0 n = 6 n = 4 n = 2 n = 0 n = −2 0.0003 0.0041 0.0304 0.1126 0.0306 0.0003 0.0037 0.0242 0.0871 0.0582 0.0006 0.0072 0.0361 0.0749 0.0360 0.0005 0.0052 0.0165 0.0277 0.0316 Intrinsic Localized Modes for gi = −5.0 n = 6 n = 4 n = 2 n = 0 n = −2 0.0006 0.0069 0.0328 0.0677 0.0328 0.0005 0.0067 0.0298 0.0609 0.0353 0.0005 0.0056 0.0202 0.0371 0.0339

n = −4 0.0042 0.0107 0.0072 0.0121

n = −6 0.0003 0.0007 0.0006 0.0012

n = −4 0.0070 0.0082 0.0109

n = −6 0.0006 0.0007 0.0010

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of intrinsic localized modes with increasing r. Again, directly to the right of the region of intrinsic localized modes is a region of multiple intrinsic localized mode solutions (i.e., for r > 1.0 and 0.9 < g < 1.0) which will be discussed elsewhere. In Table 2 the intensity profiles (i.e., λ|En,0 |2 versus n = 6, 4, 2, 0, -2, -4, -6) are presented for intrinsic localized modes at various (r, g) shown in Fig. 5. Representative solutions for all of the three branches of intrinsic localized modes are given in the Table. In general, the intensity profiles show an intensity peak near the center of the barrier that decreases rapidly to the edges of the barrier. This is a characteristic of the intrinsic localized modes. The intensity of the modes near the center of the barrier are great enough to impact the dielectric properties of the barrier media so that the modes are self-consistently created by and then sustain the dielectric properties of the barrier needed for the modes’ existence. From the results in Table 2, it is seen that, for a given value of r, modes with increasing values of g tend to exhibit intensity peaks that are broader within the barrier. For a comparison in Table 2 results are presented both for gi = 5.0 and gi = −5.0. The general features of the mode intensity plots are similar for both of these cases. The effects on the intrinsic localized modes of changing the value of gi are generally small. This is seen in the results presented in Fig. 6 which shows a plot of the g at which intrinsic localized modes occur as a function of gi for fixed r = 0.4 (denoted in the figures by the lower +’s) and r = 0.6 (denoted in the figures by the upper x’s). A mild change in the value of g is found for changing gi at a given fixed value of r. For some values of gi further splittings of the intrinsic localized mode branches are found, but these splittings are small.

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In regards to possible device applications, an interesting feature of of the plot in Fig. 5 is the presence of a set of intrinsics localized mode branches that cross each other in (r, g) space. These crossings allow for the possibility of a complex series of switchings within these systems. Moving on a circular path centered about the point of crossing of two branches of intrinsic localized modes, a series of four transmission resonances are encountered for small changes in r and g as one circulates around the crossing point of the two branches of modes. The changes in r are obtained by altering the field intensity of the guided modes whereas the changes in g may be mediated by mechanical or chemical stimuli applied to the system.

2.3.

Kerr Media Barrier with Non-Linear Off-Channel Feature

A further generalization of the off-channel impurity problem is to have the off-channel site in Fig. 4a be formed of Kerr media impurity material. In this generalization Eqs. (9) and (10) become E0,0 = g(1 + λ|E0,0 |2 )E0,0 + gl b(E−1,0 + E1,0 ) + gi b(1 + λ|E0,−1 |2 )E0,−1 (11) and

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E0,−1 = gi (1 + λ|E0,−1 |2 )E0,−1 + gb(1 + λ|E0,0 |2 )E0,0 ,

(12)

respectively. Whereas the linear off-channel site was characterized by gi , the nonlinear site is characterized by gi and λ. In Fig. 7 the transmission resonances for T > 0.6 are plotted in (r, g) space for the parameters g, k, b, gi , and λx2 used in Fig. 5. Note that the parameter λx2 for the barrier and impurity Kerr sites, for simplicity, are taken to be the same. The results in Fig. 7 are found to differ very little from those in Fig. 5. The reason for this is explained by the results for the field intensities in the barrier and off-channel sites given in Table 3. The off-channel site has, in general, a small field intensity so that its nonlinearity has little effect on the fields in the barrier sites. As a result the fields within the barrier sites are essentially the same as those in Table 2 for the off-channel site of linear dielectric media. This is found to be the case for 1.0 < gi < 5.0 and −5.0 < gi < −1.0 which have been investigated by us.

2.4.

Kerr Media Barrier with a Kerr Media or a Linear Media Impurity at the Center Site of the Barrier

Another generalization of the Kerr media barrier problem is to replace the Kerr site at the center of the barrier by an impurity site formed of a different Kerr medium. A schematic drawing of this case is given in Fig. 4b. The equations for this geometry are the same as those in Eqs. (1) through (5) with the following modifications: In Eq. (2), the equation for n = 0 is replaced by E0,0 = gi (1 + λ|E0,0 |2 )E0,0 + gl b[E1,0 + E−1,0 ]

(13)

where gi 6= g is the parameterization for the Kerr impurity site, and for simplicity we retain the same parameter λ for both the Kerr barrier sites and the impurity site at the center of the Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

164

A.R. McGurn 1

(a)

g

0.98

0.96

0.94 1

2

3

4

5

gi 1 (b)

g

0.98

0.96

0.94 -5.5

-4.5

-3.5

-2.5

-1.5

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gi Figure 6. Plots of g versus gi for the Kerr barrier coupled to a single off-channel linear media site as in Fig. 4a. In a) results are for an off-channel site formed of media with a positive gi , and in b) results are for an off-channel site formed of media with a negative gi . The upper x’s are for the r = 0.6 case and the lower +’s are for the r = 0.4 case. Aside from the values of gi , the values of parameters used in this plot are the same as those in Fig. 3. barrier. In Eq. (3), the equation for n = 0 is replaced by E1,0 = gl E1,0 + gb(1 + λ|E2,0 )|2 )E2,0 + gi b(1 + λ|E0,0 |2 )E0,0

(14)

and the equation for n = −1 is replaced by E−1,0 = gl E−1,0 + gi b(1 + λ|E0,0 |2 )E0,0 + gb(1 + λ|E−2,0 |2 )E−2,0 .

(15)

In Fig. 8a a plot is given of the resonant transmission peaks in (r, g) space with T > 0.6, using the same parameters for g, k, λx2 as in Fig. 3 and taking gi = 1.0. Again a series of Fabry-Perot modes beginning at r = 0 and extending across the figure are observed in the region 1.0 < g < 1.2. For this system, however, two different regions of intrinsic localized modes are observed. One is found below the Fabry-Perot modes (i.e., for g < 1.0 and 0.0 < r < 0.6) and one above the Fabry-Perot modes (i.e., for g > 1.2 and r > 0.2).

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Table 3. Listing for the barrier with coupling to an off-channel Kerr media site of the intrinsic localized mode field intensities. Valves are shown for λ|En,0 |2 in the barrier sites for n = 6, 4, 2, 0, -2, -4, -6 and λ|E0,−1 |2 of the off-channel site (denoted by ”off” in the Table) for selected values of (r, g) and gi for k = 2.9 and λx2 = 0.001. (r, g) (0.4, 0.9519) (0.4, 0.9593) (0.6, 0.9683) (0.6, 0.9688) (0.6, 0.9897)

n=6 0.0003 0.0003 0.0006 0.0006 0.0005

(r, g) (0.6, 0.9711) (0.6, 0.9739) (0.6, 0.9846)

n=6 0.0006 0.0005 0.0005

Intrinsic Localized Modes for gi = 5.0 n=4 n=2 n = 0 n = −2 n = −4 0.0041 0.0304 0.1126 0.0306 0.0042 0.0037 0.0242 0.0871 0.0582 0.0107 0.0072 0.0360 0.0746 0.0361 0.0073 0.0072 0.0354 0.0734 0.0368 0.0075 0.0052 0.0165 0.0277 0.0316 0.0121 Intrinsic Localized Modes for gi = −5.0 n=4 n=2 n = 0 n = −2 n = −4 0.0069 0.0328 0.0677 0.0328 0.0070 0.0067 0.0298 0.0609 0.0353 0.0082 0.0056 0.0202 0.0371 0.0339 0.0109

n = −6 0.0003 0.0007 0.0006 0.0006 0.0012

off 0.00006 0.00004 0.00004 0.00004 0.00001

n = −6 0.0006 0.0007 0.0010

off 0.00002 0.0001 0.00001

g

1.2

1

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ILM

0.8 0

0.5

1

1.5

2

r Figure 7. Plot of the resonant transmission peaks in (r, g) space for the case of a single off-channel site with gi = 5.0 and λx2 = 0.001. The results show little difference from the λx2 = 0.0, linear impurity media, results in Fig. 5. The notation ILM has been added at the lower right of the region of the intrinsic localized modes. As with the previous system studied, the lines of intrinsic localized modes begin at some r > 0 and continue with increasing r into the plot. The lowest Fabry-Perot branch meets the lower ridge of intrinsic localized modes at r ≈ 0.6, while the upper branch extends across the figure. In Table 4 results are shown for the intensity profiles within the barrier of modes within the two branches of intrinsic localized modes. It is interesting to compare the problem of a Kerr nonlinear impurity at the center of the barrier with the case in which the center site barrier impurity is made of linear impurity media. For this case, Eq. (13) becomes E0,0 = gi E0,0 + gl b[E1,0 + E−1,0 ],

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(16)

166

A.R. McGurn 1.4 ILM

(a)

g

1.2

1 ILM

0.8 0

0.5

1

1.5

2

r 1.4 (b)

g

1.2

1 ILM 0.8 0

0.5

1

1.5

2

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r Figure 8. Plot of the resonant transmission peaks in (r, g) space for the case of a Kerr barrier containing an impurity site at it center: a) Kerr impurity site with gi = 1.0 and λx2 = 0.001. The notation ILM has been added at the lower right of the g < 1.0 region of the intrinsic localized modes and above and adjacent to the g > 1.2 region of intrinsic localized modes. b) A linear media impurity site with gi = 1.0. The notation ILM has been added to the lower right of the g < 1.0 region of intrinsic localized modes.

Eq. (14) becomes E1,0 = gl E1,0 + gb(1 + λ|E2,0 |2 )E2,0 + gi bE0,0 ,

(17)

and Eq. (15) becomes E−1,0 = gl E−1,0 + gi bE0,0 + gb(1 + λ|E−2,0 |2 )E−2,0 .

(18)

This amounts to taking λ = 0 at the center site, but λ 6= 0 on the other Kerr sites of the barrier. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

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Table 4. Listing of the intrinsic localized mode field intensities for the barrier with a Kerr impurity site at the center of the barrier, i.e., at (0, 0). Valves are shown for λ|En,0 |2 in the barrier sites for n = 6, 4, 2, 0, -2, -4, -6 for selected values of (r, g) and for k = 2.9 and λx2 = 0.001. The impurity site has gi = 1.0 and λx2 = 0.001.

(r, g) (0.3, 0.9634) (0.4, 0.9750) (0.5, 0.9850) (0.6, 1.2972) (1.0, 1.2684)

n=6 0.0001 0.0002 0.0004 0.0003 0.0009

Intrinsic Localized Modes n = 4 n = 2 n = 0 n = −2 0.0020 0.0125 0.0506 0.0125 0.0029 0.0142 0.0415 0.0141 0.0039 0.0145 0.0308 0.0146 0.0016 0.0117 0.1362 0.0118 0.0033 0.0167 0.1416 0.0163

n = −4 0.0020 0.0029 0.0039 0.00176 0.0027

n = −6 0.0002 0.0002 0.0004 0.0005 0.0003

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In Fig. 8b results are presented for the gi = 1.0 linear impurity and the usual values of g, b for k = 2.9 and λx2 = 0.001. The results are similar to those in Fig. 8a, with a renormalization of the the plotted ridges for g < 1.2. For g > 1.2, however, the branch of intrinsic localized modes found in Fig. 8a is no longer present in the results of Fig. 8b. This indicates the importance of the nonlinear media at the center site in developing intrinsic localized mode excitations. This is due to the fact that the mode intensity is greatest at the center site so that the dielectric nonlinearity exhibits the largest response at the center site. In addition to the renormalization of the lines and ridges for g < 1.2 a new branch of intrinsic localized modes is found that develops near horizontally with increasing r.

3.

Conclusions

The properties of a number of simple photonic crystal circuit elements containing a barrier of Kerr nonlinear media and various linear media and nonlinear media impurities have been discussed. The possible modes that can be resonantly excited by guided waves incident on the barriers have been determined and listed in an (r, g) space plot of the transmission resonances of the barrier. This is a type of spectroscopy which allows for the modes of the barrier to be classified and their behaviors under changes in the nonlinear barrier media shown. Emphasis was placed on the determination of the Fabry-Perot and intrinsic localized modes. While the Fabry-Perot modes exist in the linear limit of the barrier media and exist in renormalized forms as the barrier nonlinearity is increased, the intrinsic localized modes only are found in the case of nonlinear barriers as they only exist in nonlinear systems. The various ridges and lines of resonances in (r, g) are such that barrier modes on the same ridge or line have similar wavefunctions whereas modes on different ridges or lines have distinctly different types of wavefunctions. This gives a very useful classification scheme for the possible modes that are excited within the barrier. The nonlinear barriers display characteristics important in various types of amplification, switching, and rectification applications. In addition, the crossing of different branches of intrinsic localized modes can be used to formulate an intricate series of switching processes that are provided for by a number of different mechanisms.

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Appendix A

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A brief summary of the origin of the difference equation formulation is given here for the case in which the electric field of the modes is polarized with the electric field parallel to the axes of the dielectric cylinders. For more details the reader is referred to Refs. [9], [10] and [25]. The photonic crystal is described by a periodic dielectric function ǫ(~x|| ) where ~x|| is in the plane of the Bravais lattice, perpendicular to the axes of the dielectric cylinders. A waveguide is defined by introducing a change in the dielectric constant, δǫ(~x|| ), in a row of dielectric cylinders. Using the Helmholtz equation for the electric fields of the modes and exact methods of Green’s functions[9, 10, 11, 25, 26], the electric fields of the waveguide modes satisfy [9, 10, 25] Z ω2 (19) d2 x′|| G(~x|| , x~′ || )δǫ(x~′ || )E(x~′ || ). E(~x|| ) = c2 Here G(~x|| , x~′ || ) is the Green’s function of the Helmholtz equations for the photonic crystal described by ǫ(~x|| ). The evaluation of the Green’s functions for the system addressed in this paper is discussed in Refs.[9, 54] where it is shown that the Green’s functions for frequencies within the photonic crystal stop band exhibit an overall rapid decay with increasing separation of ~x|| and x~′ || within the system, particularly for multiple separations of the lattice constant[23]. The stop band decay is due to the non- propagating nature of the wave functions for frequencies within the stop band, and is different from the behavior shown at pass band frequencies. If δǫ(~x|| ) is non-zero only in a small region about the axes of cylinders forming the waveguide channel and E(~x|| ) changes slowly over δǫ(~x|| ) in each such cylinder, then Eq. (19) reduces (for a waveguide along the x−axis for which only on site and nearest neighbor site interactions are significant) to a set of difference equations given by En,0 = gp [fn,0 En,0 + b(fn+1,0 En+1,0 + fn−1,0 En−1,0 )].

(20)

In Eq. (20), fn,0 = 1 + λ|En,0 |2 with λ = 0 for linear dielectric media, gp = R 2 ′ ω2 d x G(0, x~′ || )δǫ(x~′ || ) where the integral is over the cylinder centered at (0, 0), and 2 c R 2|| ′ R b = d x|| G(a0ˆi, x~′ || )δǫ(x~′ || )/ d2 x′|| G(0, x~′ || )δǫ(x~′ || ) where the integrals are computed as in the case of gp and a0 is the lattice constant of the waveguide. In the evaluation of all of the integrals in the definitions of gp and b, λ = 0. Both the Green’s functions and their space integrals are evaluated from the eigenvalues and eigenvectors of the electromagnetic equations of motion for the photonic crystal[9, 54] using methods that are common in the study of ionic impurities in metals[55]. The parameters of the photonic crystal used in this paper are the same as those used in a number of previous publications, and we refer the read to these for a more detailed discussion[9, 10, 11, 25]. The two-dimensional photonic crystal is a square lattice with cylinders of dielectric constant ǫ = 9 and radius R = 0.37796ac where ac is the lattice constant of the square lattice, and waveguide and barrier impurity materials occur in a square cross section of side 0.02ac . The waveguide is taken along the x− axis of the square lattice such that the lattice constant of the waveguide is a0 = ac . The dielectric

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constant of the Kerr media is taken of a generic isotropic form so that a semi-quantitative indication of the effects of nonlinearity are given. This is not an uncommon thing to do in theoretical treatments in nonlinear optics, and it is hoped that the results presented will stimulate studies on specific systems in which more detailed considerations of the form of the dielectric constants may be made.

References [1] Pollaock, C.; Lipson, M. ”Integrated Photonics”, Kluwer Academic Publishers: Boston, 2003. [2] Prasad, P. N. ”Nanophotonics”, John Wiley and Sons: Hoboken, 2004. [3] McGurn, A. R. In ”Survey of Semiconductor Physics”; Boer, K. W.; Ed.: John Wiley and Sons, Inc.: New York, 2002, Chp. 33. [4] Sakoda, K. ”Optical Properties of Photonic Crystals”; Springer: Berlin, 2001. [5] Joannopoulos, J. D.; Meade, R. D.; Winn J. N. ”Photonic Crystals”; Princeton University Press: Princeton, 1995. [6] Joannopoulos, J. D.; Vilenueve, P. R., Fan, S. Nature 386, 1997, pp. 143-149.

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[7] Londergan, J. T.; Carini, J. P.; Murdock, D. P. ”Binding and Scattering in TwoDimensional Systems: Applications to Quantum Wires, Waveguides, and Photonic Crystals”; Springer: Berlin, 1999. [8] Zolla, F.; Renversez, G.; Nicolet, A.; Kuhlmey, B.; Guenneau, S.; Felbacq, D. ”Foundations of Photonic Crystal Fibers”; World Scientific: New Jersey, 2005. [9] McGurn, A. R. Phys. Rev. B 53, 1996, pp. 7059-7064. [10] McGurn, A. R. Phys. Rev. B 61, 2000, pp. 13235-13249. [11] McGurn, A. R. Phys. Rev. B 65, 2002, pp. 75406-1 - 75406-11. [12] Hunsperger, R. G. ”Integrated Optics: Theory and Technology”, 4th Ed. Springer: Berlin, 1984. [13] Yeh, C. ”Applied Photonics”. Academic Press: San Diego, 1990 Chps. 2 and 12. [14] Abdonovic, I.; Uttamchandarri, D. ”Principles of Modern Optical Systems”, Artech House: Norwood, 1989. [15] Tanida, J.; Ichioka, Y. In ”Progress in Optics”; Wolf, E.; Ed.; Elsevier: Amsterdam, 2000, Vol. XL, pp 77-114. [16] Kilin, S. Ya. In ”Progreess in Optics”, Vol. 42, E. Wolf, E.; Ed.; Elsevier: Amsterdam, 2001, Vol. 42, pp. 1-92. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

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[17] Glesk, I.; Wang, B.C.; Xu, L.; Baby, V.; Prucnal, P. R. In ”Progress in Optics”; Wolf, E.; Ed.; Elsevier: Amsterdam, 2003, Vol. 45, pp. 53-118. [18] Feitelson, D. G. ”Optical Computing”; MIT Press: Cambridge, 1988. [19] ”Optical Computing”; Wherrett, B. S.; Tooley, F. A. P.; Ed.; Edinburgh University Press: Edinburgh, 1989. [20] Mansuripur, M. ”Classical Optics and Its Applications”; Cambridge University Press: Cambridge, 2002. [21] Kivshar, Y. S.; Agrawal, G. P. ”Optical Solitons”; Academic Press: Amsterdam, 2003. [22] Ibach, H.; Luth, H. ”Solid-State Physics”; Springer: Berlin, 2003. [23] McGurn, A. R. Chaos 13, 2003, pp. 754-765. [24] McGurn, A. R. J. Phys. CM 14, 2004, pp. S5243-S5252. [25] McGurn, A. R. Phys. Lett. A 251, 1999, pp. 322-335. [26] McGurn, A. R. Phys. Lett. A 260, 1999, pp. 314-321. [27] McGurn, A. R.; Birkok, G. Phys. Rev. B 69, 2004, pp. 235105-1 - 235105-13. [28] Cowan, A. R.; Young, J. F. Phys. Rev. E 68, 2003, pp. 46606-1 - 46606-16.

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[29] Yanik, M. F., Fan, S., Soljacic, M., and Joannoupoulis, J. D., Opt. Lett. 28, 2003, pp. 2506-2510. [30] Soljacic, M.; Joannopoulos, J. M. Nat. Mater. 3, 2004, pp 211,215. [31] Xu, Z.; Maes, B.; Jeang X.; Joannopoulos, J. D.; Torner, L.; Soljacic, M. Optics Letters 33, 2008, 1763-1767. [32] Mills, D. L. ”Nonlinear Optics”; Springer-Verlag: Berlin, 1998. [33] Banerjee, P. P. ”Nonlinear Optics”; Marcel Kekker, Inc.: New York, 2004. [34] Boyd, R. W. ”Nonlinear Optics 2nd Ed.”; Academic Press: Amsterdam, 2003. [35] Kittel, C. ”Introduction to Solid State Physics”; Wiley: New York, 1976. [36] Berger, V. Phys. Rev. Lett. 81, pp. 4136-4139. [37] Shi, B.; Jiang, Z. M.; Wang, X. Opt. Lett. 26, 2001, pp. 1194-1196. [38] Ren, F. F.; Li, R.; Cheng, C.; Wang, H. T.; Qui, J. R.; Si, J. H.; Hirao, K. Phys. Rev. B 70, 2004, pp. 245109-245113. [39] McGurn, A. R. Phys Rev B 27, 2008, pp. 115105-115115. [40] McGurn, A. R. J Phys: Condens. Matter 20, 2008, 025202-025212. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

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[41] McGurn, A R. in ”Nonlinear Research Perspectives” Ed. Wang, 2007, Novascience Publishers. [42] McGurn, A. R. J Phys: Condes. Matter 21, 2009, pp.485302- 485312. [43] McGurn, A. R. Compexity 12, 2007, pp.18-40. 39 [44] Merzbacher, E. ”Quantum Mechanics”, 3rd Edition, 1998, Hamilton Printing Company, New York. [45] Sievers, A, J.; Takeno, S. Phys. Rev. Lett. 61, 1988, pp. 970-974. [46] Sievers, A. J.; Page, J. B. In ”Dynamical Properties of Solids”; Horton, G. K.; Maradudin, A. A.; Ed.; Elsevier: Amsterdam, 1995, Vol. 7, pp. 137-255. [47] Lam L. (ed.) ”introduction to Nonlinear Physics”. 1996 Springer, New York [48] Pau, T. ”Attractors, Bifurcations, and Chaos”, 2000, Springer, New York [49] Rasband, S. N. ”Chaotic Dynamics of Nonlinear Systems” 1990, Wiley, New York. [50] Turing, A. M., Phil. Trans. R. Soc. B 237, 1952, 37-55. [51] Rabinovich, M. I.,; Ezershy, A. B.; Weidman, P. D. ”The Dynamics of Patterns” 2000, World Scientific, Singapore. [52] Wolfram, S. ”A New Kind of Science”, 2002, Wolfram Media, Urbana.

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[53] McGurn, A. SPIE Proceedings, Vol. 7395, 2009, Ed. by E. S. Kawata, V. M. Shalaev, and D. P. Tsai, 7395 1T-1 to -11. [54] Maradudin, A. A.; McGurn, A. R. J Opt. Soc. Am. B 10 pp. 307-316. [55] Callaway, J. ”Energy Band Theory”, 1963, Academic, New York.

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In: Photonic Crystals Editor: Venla E. Laine, pp. 173-189

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Chapter 8

INFLUENCE OF THE SOLID-LIQUID ADHESION ON THE SELF-ASSEMBLING PROPERTIES OF COLLOIDAL PARTICLES Edina Rusen1, Alexandra Mocanu1, Bogdan Marculescu1, Corina Andronescu1, I.C. Stancu1, L.M. Butac1, A.Ioncea2 and I. Antoniac3

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1

University Politehnica Bucharest, Depart. of Polymer Science, Bucharest, Romania 2 S.C. Metav S.A. Bucharest,C.A., Bucharest, Romania 3 University Politehnica Bucharest, Faculty of Materials Science and Engineering, Bucharest, Romania

Abstract Colloidal dispersions with self-assembling properties have been prepared by soap-free emulsion copolymerization of styrene (St) with hydroxyethylmethacrylate (HEMA), using various St/HEMA molar ratios. Investigations by SEM, AFM and reflexion spectra have shown that copolymer particles give colloidal crystals while monodisperse particles of polystyrene, with no HEMA units, do not crystallize. XPS analysis of the copolymer particles has shown that the HEMA hydrophilic units tend to concentrate on the surface of the copolymer particles, while the core is richer in St. The copolymer particles have been separated and redispersed in other liquids than water (ethylene glycol, formamide, ethanol, hexane, and toluene) to investigate the conditions that allow self-assembling after the diluent evaporation. For each case, contact angles have been measured and the adhesion work has been computed. The results have shown that the contact angle does not influence the selfassembling process, while the adhesion work is an important parameter. Only dispersions characterized by a value of the adhesion work superior to 87 mJ/m2 did crystallize after the diluent evaporation.

Keywords: soap-free polymerization; self-assembling; contact angle; adhesion work

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1. Introduction Spontaneous formation of photonic crystals from colloidal dispersions of either organic polymers or inorganic materials has raised a great scientific interest materialized in a large number of papers published in the last decade. Although many alternative methods for obtaining colloidal crystals by deposing spherical, monodisperse particles, on plane surface have been reported [1,2], the autoassembling process always involves three distinctive stages: • •

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•

a concentration of the colloidal particles on the substrate surface, phenomenon based on the reciprocal adherence; a compact disposition of the particles in a planar 2D network due to the lateral inter-particles attractions; an increase of the thickness of the layer by successive deposition of new crystalline planes on the initial planar area, thus leading to a 3D – spatial network, with a compact hexagonal (hcp) or centered cubic (ccp) geometry.

Since none of the above stages has been proven as dominant for the entire process, it is reasonable to accept that the relative importance of the stages is the same as their chronological order. One of the major problems is to define the nature of the motrix force of the auto-assembling process. An exhaustive and pertinent analysis of the complex physical interaction phenomena involved during concentration of the disperse particles [3,4] revealed the fact that the capillary forces play a decisive role, both due to their intensity and their interaction radius. Consequently, most of the theoretical studies focus on obtaining precise methods for computing the capillary force, based mainly on the geometry of the physical system and less on the chemical nature of the solid surface or the liquid medium [5-21]. The only accessible parameter (by experimental measurements) that reflects the specifics of these interactions is the contact angle (θ) between the liquid and the solid surface. The most relevant results from the above-mentioned studies lead to some conclusions that are considered relevant premises for the present work: a.

Several theoretical studies are oriented towards obtaining mathematical models describing physical systems consisting of pairs of solid bodies separated by a liquid film. The two bodies have well defined shape and size, but their chemical nature is ignored. The chemical structure of the liquid is not involved either and it's only physical property taken into account is the superficial tension (γ) [5-12]. In other works, where experimental data is used to validate the theoretical models proposed, the liquid involved is always water and the dispersed solid phase is composed by monodisperse particles of polystyrene, SiO2 [1,2,16,18,20] or submicronic sand particles [21]. As for the plane, solid surfaces used as the support for the deposition and the spontaneous auto-assembling of the particles, glass (as such or with the surface covered by hydrophobic films of perfluorderivatives or organo-siloxane compounds [2,18]), Au, Ag, mica [16] were used. A notable exception is the study of the auto-aggregation of ice or

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tetrahydrofurane hydrate from n-dodecane [15]. Note that the attraction resulted from capillary forces acts on any particle, whatever the nature and shape of the dispersed particles but the result is (or may be) a crystalline network only if these are spherical and with a strictly uniform size. b. Many works assume explicitly (even if only qualitatively) that the contact angle has a crucial influence on the attractive forces and consequently on the autoassembling capacity of the colloidal particles [13,14,16,20]. This is obviously due to the decisive role of the interfacial compatibility between the solid and the liquid. The shape of the liquid meniscus (concave or convex), the value and the sign of the capillary force depend on the interfacial compatibility. The role played by the contact angle is explicitly or implicitly considered in more complex mathematical models [9,16,18]. Moreover, the relations used to compute the capillary force used in most studies [11,12,15,17,19] are those given by equations (1) and (2) below: Fc∼γ (cos θ1+ cos θ2)

(1)

Fc∼γ cos θ

(2)

Relation (1) applies for attractions between surfaces of different chemical composition (S1 and S2) separated by a liquid film with the superficial tension γ. Relation (2) is valid for the capillary attraction between two spherical surfaces having the same nature. Obviously, relation (1) reduces to (2) for two surfaces with the same chemical composition (θ1 = θ2). For two different surfaces, from which one (S2) has a strong liophilic character (therefore the liquid spreads on the respective surface) (θ < 100 leads to cos θ ≈ 1), relation (1) reduces to equation (3): Fc∼γ (cos θ1+ 1) c.

(3)

Many papers use the term "adhesive forces" instead of "capillary forces" and replace the term "attraction" with "adhesion". The use of such notions in works of a high scientific level is not figurative but reflects a real phenomenon: the liquid film that mediate the attraction between the surfaces of two solid bodies acts as an adhesive layer that connects two identical or different substrates. If the liquid (the adhesive) is superficially compatible with the solid surface (θ < 900 leading to cos θ > 0), then the capillary forces are attractive, while for an incompatibility between the liquid and the solid (θ > 900 leading to cos θ < 0) the same forces are repulsive. Meanwhile, considering the capillary force as derivated from an attraction potential W, gives an alternative definition to the classical one (Laplace) [5,6]:

Fc = −

dW dL

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(4)

176

Edina Rusen, Alexandra Mocanu, Bogdan Marculescu et al. Even if the physical nature of this potential is not clearly defined, obviously it must be an electrostatic potential, the same that is considered as the basis for the intermolecular attractive forces (dispersion, orientation and induction forces). In fact the energetic approach of any adhesive assembling results form the interpenetration of the hybrid force fields localized at the solid-liquid interface (adhesion) and the homogeneous forces acting in the bulk of the liquid film (cohesion).

Therefore, the auto-assembling capacity of the colloidal particles should take into account not only the capillary forces but also the adhesion work (Wa), a thermodynamic parameter used frequently to characterize the interfacial compatibility between a liquid and a solid. Wa may be obtained from experimental data using the classical relation Young-Dupré:

Wa = γ L (1 + cos θ)

(5)

Note also that from the above relations it results that Fc ≈ Wa. Using the adhesion work makes possible not only the interpretation of the experimental data already available but also it allows predicting the behavior of a solid-liquid system by using analytical relations such as Wa = Wa ( γS , γS , γ L, γ L ) , proposed by Fowkes or Wu [22,23]. The necessary and sufficient condition for using such equations is to know the values of the dispersive and respectively polar components of the superficial tensions involved in the adhesion phenomenon. The aim of the present work is to validate the above hypothesis. Colloidal dispersions of various chemical compositions have been obtained and the behavior of the particles in a variety of liquids, with different chemical composition, has been examined. The investigation supposes measuring, for every solid-liquid pair, of the adhesion work, coupled with evaluating the aptitude of crystallization of the same systems on a single plane substrate (hydrophilic glass). Obviously, the selected colloidal suspensions must have a size monodispersity, since this is the most important condition for self-assembling. The disperse particles are styrene (St) – hydroxyethylmethacrylate (HEMA) copolymers with various compositions while liquids used have been selected according to both their superficial tensions and wide range of polarities.

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D

P

D P

2. Materials and Methods 2.1. Materials The St purchased from Merck has been purified through distillation under vacuum. HEMA (Aldrich) has been passed through separation columns filled with Al2O3 to remove inhibitors. The initiator used, potassium persulphate (KPS) (Merck) has been recrystalized from a mixture ethanol/water and then vacuum dried. For the precipitant homopolymerisation of St the diluent was methanol (M) (Merk) previously purified by distillations. The initiator was azo-iso-butyro-di-nitrile (AIBN) (Merk) purified by recrystallisation from M.

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For redispersing monodisperse particles and respectively for measuring contact angles several liquids have been used: formamide (Merck), ethylene-glycol (Aldrich), ethanol (Aldrich), hexane (Fluka) and toluene (T) (Aldrich).

2.2. Soap-Free Emulsion Polymerization A total mass of 1.55 g of monomer mixture (with a different molar ratio St:HEMA for each test) has been added to 20 ml distilled water together with a constant concentration of KPS (2.3⋅10-3 mol/l). The reaction mixture was nitrogen bubbled and then maintained for 48 h at 75°C under continuous stirring. In test A the monomer was only styrene while tests B, C, D and E consisted of mixtures St:HEMA with the molar ratios 17:1, 8:1, 5:1 sand 3.3:1 respectively. The final lattices have been dialyzed in distilled water for 7 days, using cellulose dialysis membranes (molecular weight cutoff: 12.000-14.000). Latexes have been precipitated in M, filtered and dried under vacuum before measuring the final conversions (that have been in all cases above 99%) and to obtain the cylinders used for measuring the contact angles.

2.3. Precipitant Polymerization

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2 mol/l St have been added to 10 ml M together with an AIBN concentration of 2⋅10-3 mol/l. After nitrogen purging the polymerization vial has been kept for 4 hors at 600C, under stirring. The resulted polymer was separated by filtration and the unreacted monomer was eliminated by Soxhlet extraction and dried under vacuum until a constant mass has been reached.

2.4. Obtaining the Colloidal Crystals The colloidal dispersions (in water or other liquids) have been deposed by dropping on lamellae of hydrophilised glass (the hydrophilic character was obtained by immersing the glass in a 30% NaOH solution for 24 hours) degreased immediately before use. Specimens (colloidal dispersions) have been maintained at constant temperature (400C) until complete drying.

2.5. Redispersing of the Monodisperse Particles in Liquid Media The opal was detached from the glass sheets with a metallic blade, manually grinded in a mortar and redispersed by ultrasounds in various liquid media, at concentrations similar to the initial latexes (1.55 g/20 ml).

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2.6. Characterization The morphologies of polymer particles were observed using a Zeiss Evo 50 XVP Scanning Electron Microscope (SEM). The probes were sputtered with a thin layer of gold prior to imaging. Structural characterizations of the opals were performed using the AFM NTMDT P47H apparatus. The reflectance has been recorded using a UV-VIS spectrophotometer Jasco V550. A XPS K-ALPHA spectrometer has been used to establish the specimen composition. Contact angles have been measured with a EW-59780-20 Contact Angle Meter 110 VAC, 50/60Hz. The method used for obtaining the specimens was different from the one used for copolymers with similar composition reported in the literature [24,25]; in the present work, solid particles have been compressed under a pressure of 15MPa using a PerkinElmer mini-press.

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3. Results and Discussion Since there is a well-known difference of solubility in aqueous media of the two monomers (St, respectively HEMA) a first stage of the research was to synthesize copolymers with various compositions by soap-free emulsion polymerization. The differences in the solubilities of the monomers, macroradicals and respectively polymers will result in differences in the size of the final particles [26,27], since the particle size is directly influenced by the mechanism of their generation and growth. Therefore, the films obtained from the latexes thus obtained have been investigated using the SEM technique to asses their morphology (figure 1). The analysis of the images in figure 1 shows that the particles size diminishes when the molar ration St/HEMA is lower. Since both the initiator concentration and the total mass of the monomers have been kept constant in all the synthesis processes, the only possible explanation for the increase in the number of particles when the St/HEMA molar ratio diminishe resides in the change of the topochemistry of the process resulted from the different concentrations of the hydrophilic (water-soluble) monomer in the reaction mixture. A reasonable supposition is that St/HEMA ratio influences the kinetics of the process both in the nucleation stage and in what concerns the subsequent behavior of the generated particles. This remark is valid for several other binary systems consisting of a hydrophobic and a hydrophilic monomer [28,29]. A comparison between the morphology of the particles from latexes A-E (from the SEM data, figure 1) shows that for the particles containing HEMA units the monodispersity allows the self-assembling leading to colloidal crystals. Supplementary investigation methods, such as AFM (figure 2) and reflexion spectra have been used to give more information on the colloidal crystals. From figure 2, a well-ordered, close packed structure can be noticed with a triangular arrangement, that can be attributed to planes of a face centered cubic (fcc) system [30,31]. Moreover FTT’s image displays sharp peaks confirming the existence of long range crystalline order.

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Figure 1. SEM images of the particles obtained from the synthesized dispersions (in water): A (d=430nm) monodisperse, B (d=380nm) monodisperse, C (d=330nm) monodisperse, D (d=290nm) monodisperse and E (d=254) monodisperse

Figure 2. AFM image of the top surface of the opal and FFT’s image from the samples B and C. Photonic Crystals: Fabrication, Band Structure and Applications : Fabrication, Band Structure and Applications, Nova Science Publishers,

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In figure 3, the reflection peaks located at 448 nm and 618 nm, are due to the Bragg reflection of periodic arrangement of spheres in the solid layer. A sharp peak indicates a regular, periodic structure.

Other works concerning the analysis of the St/HEMA soap-free copolymerization have shown a core-shell morphology of the colloidal particles [24,25]. Several characterization methods have been used to prove that the superficial layer is richer in HEMA as compared to the global composition of the copolymer. Since in the present work the superficial behavior of the particles generated from the St/HEMA system is investigated, a more precise characterization of the chemical composition of the superficial layer has been obtained by XPS. E D C B A

4

3

nr. of level

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Figure 3. Reflection spectra of the opal films from the systems B and C.

2

1

0 0

10

20

30

40

50

60

70

80

90

100

% St (molar) Figure 4. XPS results for depths between 0 and 160 nm (the shaded area gives the limits of the ST fraction from the initial composition of the monomers

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St HEMA

4

3

nr. of level

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Since the nucleation is homogeneous [32-35], the macroradicals precipitated during this stage will be found on the surface of the final particles; their composition and molar mass (depending on the initial ratio of the monomers) are important factors acting on the size of the nuclei at the moment when colloidal stability is achieved (at the end of the nucleation stage). Obviously, the colloidal stability of the particles is given both by the ionic groups provided by the initiator and the hydroxyl, water-soluble groups from the HEMA units. The XPS analysis gave the composition of the particles obtained from latexes A-E both on the surface and for depths reaching up to 160 nm (which corresponds roughly with the center of a particle); scannings were performed every 40 nm (figure 4). For system A the surface contains 95% polystyrene (PSt) while the rest consists of oxygen and sulphur, constituents of the SO4- groups resulted from the initiator. In all the other specimens, the percentage of PSt in the superficial layer of the particles is much lower than the corresponding fraction of St in the initial monomer mixture. This is another argument towards the homogeneous nucleation. A more important remark is that the PSt percentage on the surface decreases with the St/HEMA ratio in the initial monomer composition [25], which confirms that the composition of the macroradicals that reach the critical precipitation size is different. Therefore, the size of the nuclei when the colloidal stability achieved is different. If the initial composition of the monomer mixture is richer in HEMA, the critical precipitation size increases, the size of the auto-stabilized nuclei is lower and therefore the final size of the particles is lower (meaning that more particles will be generated). When the analysis is made at various distances from the surface of the particles the content of St increases, up to values higher than the St content in the initial monomer mixture, thus balancing the superficial lower amount of the same monomer. To reduce the level of error, sample E has been re-analyzed, with a step of 3 nm. Results are shown in figure 5.

2

1

0 0

10

20

30

40

50

60

70

80

90

molar % of monomers Figure 5. XPS analysis of specimen E with a step of 3 nm.

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Influence of the Solid-Liquid Adhesion…

183

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Figure 5 also shows a slight decrease of the HEMA percentage with the distance from the surface, which confirms the accuracy of the data previously obtained. To confirm the results, specimen B was dissolved in N-methyl pyrrolidone and then deposed on a glass sheet. After solvent evaporation the transparent layer has been analyzed. XPS data (figure 6) showed no difference between the results at different depths and the copolymer composition was identical with the one of the initial monomer mixture.

Figure 6. XPS analysis for the film resulted from specimen B after dissolution in NMP and evaporation of the solvent.

The results (figure 4-5) show clearly that the HEMA units are concentrated in the superficial layer of the particles, thus increasing the hydrophilicity above the value corresponding to the global composition of the copolymers. This result is valid for all the initial compositions of the monomer mixture. Since the superficial roughness has a strong influence on the lipophilicity (and consequently on the values of the contact angle) a characterization of the surfaces microgeometry (by AFM) was considered as mandatory. Figure 7 shows the surface of the particles before (a) and after (b) pressure was applied. Case (b) shows a significant lower surface roughness (which should give relevant data resulting from the contact angle measurement) and the details show a change of shape of the initially spherical particles, which confirms that the diminished roughness is not accompanied by a change in the chemical nature of the surface.

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a)

b) Figure 7. AFM images of the particles surface before and after pressing.

An initial validation of the method was performed by measuring the contact angle of water on PSt, that do not contain any ionic, hydrophilic groups. The polymer was obtained by precipitant polymerization in M, with AIBN as initiator. Fine particles (d=0.1-1 μm) have been obtained, dried and then pressed into cylinder, using the same technique as for the products of soap-free emulsion polymerization. The measured contact angle, θ = 94.59±20,

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Influence of the Solid-Liquid Adhesion…

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was practically identical with the theoretical value. The theoretical value, θ = 920, was obtained by computation, using the relation: γL(1+cosθ) = 2

(

LW γ LW + γ L+ γS− + γ −L γS+ L γS

)

where L:H2O, S:PSt and :γH2O = 72.8 mN (γLW = 21.8; γ+ = γ- = 25.5), γPSt = 40.7 mN/m (γLW=42; γ+=0; γ-=1.1) [23]. If the value of the adhesion work would determine the nucleation and growth of the colloidal crystals, then one could anticipate that the spherical monodisperse particles should self-assemble also from other liquids than water. The self-assembling condition should be that the value of the adhesion work must exceed a given threshold value, decided by the contact angle of the liquid on the solid surface and the superficial tension of the liquid. Both the adhesion work (computed according to the Young- Dupré relation) and the term γcosθ (which decides the value of the capillary force) increase with the polarity of the liquids, for the same solid surface (constant γSP). Reciprocally, for a given liquid (a constant γLP) the behavior towards the solid surface is decided by the polarity of the solid, i.e. by the St/HEMA ratio in the copolymer. Therefore, the copolymers A-E have been colloidally dispersed in ethylene-glycol, formamide, ethanol, hexane, toluene and then deposed on hydrophilic glass lamellae, to examine the possible tendency towards self-assembling. For all cases, the contact angles of the same liquids on the surfaces of the particles obtained by copolymerization have been measured (table 1). The data in table 1 lead to some interesting conclusions:

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•

•

•

The self-assembling capacity was first evaluated by the apparition of the iridescence characteristic for opal and subsequently proved by SEM investigation. For all specimens that have shown selective diffraction the SEM images revealed a colloidal crystal structure while the specimens with a whiteopaque aspect have shown a non-ordered morphology (similar to specimen A from figure 1); The value of the contact angle by itself does not influence the self-assembling process. Even if the surface has been perfectly wetted (θ